Notation by work
Paper VII: Quantum Seed Calculus
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1144 symbol spellings in 2761 mathematical passages across 53 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
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*Definition passage · 272 passages
Search aliases: *
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 54
inition:001 A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
H=H^*
+Definition passage · 164 passages
Search aliases: +
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2345
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
R_\ell:\Hcal_\ell\longrightarrow\Hcal_{\ell+1},
=Definition passage · 1076 passages
Search aliases: =
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
ADefinition passage · 133 passages
Search aliases: A
Definition passage: Finite quantum seed system · Read passage · 02_strengthened.tex, line 93
[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.
Exact mathematical passage
\Acal S = \operatorname{span}\{A\psi:A\in\Acal,\ \psi\in S\}.
aDefinition passage · 379 passages
Search aliases: a
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 944
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
(P_a)
\absDefinition passage · 26 passages
Search aliases: abs
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\AcalDefinition passage · 12 passages
Search aliases: Acal A mathcal
Definition passage: Finite quantum seed system · Read passage · 02_strengthened.tex, line 89
[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.
Exact mathematical passage
(\Hcal,\Acal,S),
BDefinition passage · 85 passages
Search aliases: B
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1502
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
\Sigma(z)=B(z-D)^{-1}B^*
bDefinition passage · 133 passages
Search aliases: b
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 945
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Pcal_{ab}(X)=P_aXP_b.
B^*Definition passage · 27 passages
Search aliases: B^*
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1502
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
\Sigma(z)=B(z-D)^{-1}B^*
\BcalDefinition passage · 1 passage
Search aliases: Bcal B mathcal
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 949
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Bcal_{ab}=P_a\End(\Hcal)P_b
\Bcal_{ab}Definition passage · 1 passage
Search aliases: Bcal_ab B_ab Bcal mathcal
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 949
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Bcal_{ab}=P_a\End(\Hcal)P_b
CDefinition passage · 182 passages
Search aliases: C
Definition passage: Universal closure obstruction · Read passage · 02_strengthened.tex, line 2186
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
\Ob_{ab}^{\rm cl} = q_{\mathfrak c}([C_a,C_b]).
\mathfrak cDefinition passage · 6 passages
Search aliases: c mathfrak
Definition passage: Universal closure obstruction · Read passage · 02_strengthened.tex, line 2182
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
q_{\mathfrak c}:\End(\Hcal)\longrightarrow\End(\Hcal)/\mathfrak c
cDefinition passage · 65 passages
Search aliases: c
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
C_aDefinition passage · 25 passages
Search aliases: C_a
Definition passage: Universal closure obstruction · Read passage · 02_strengthened.tex, line 2186
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
\Ob_{ab}^{\rm cl} = q_{\mathfrak c}([C_a,C_b]).
C_bDefinition passage · 11 passages
Search aliases: C_b
Definition passage: Universal closure obstruction · Read passage · 02_strengthened.tex, line 2186
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
\Ob_{ab}^{\rm cl} = q_{\mathfrak c}([C_a,C_b]).
C_cDefinition passage · 7 passages
Search aliases: C_c
Definition passage: Anomaly free continuum constraint system · Read passage · 02_strengthened.tex, line 2540
[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotient or physical subspace.
Exact mathematical passage
[C_a,C_b]\psi = \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
C_{\ell,r}Definition passage · 1 passage
Search aliases: C_ℓ,r ell
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
C_{\ell,r}
\CCDefinition passage · 41 passages
Search aliases: CC C mathbb
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
\CcalDefinition passage · 4 passages
Search aliases: Ccal C mathcal
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
\cHDefinition passage · 18 passages
Search aliases: cH H mathcal
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\cODefinition passage · 5 passages
Search aliases: cO O mathcal
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\cRDefinition passage · 6 passages
Search aliases: cR R mathcal
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
DDefinition passage · 90 passages
Search aliases: D
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1501
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
z\notin\Spec D
dDefinition passage · 147 passages
Search aliases: d
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1961
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
d_L(t) =-2\frac{\dd\log Z_L(t)}{\dd\log t} = 2t \frac{\Tr(L\e^{-tL})}{\Tr(\e^{-tL})}.
d_LDefinition passage · 5 passages
Search aliases: d_L
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1961
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
d_L(t) =-2\frac{\dd\log Z_L(t)}{\dd\log t} = 2t \frac{\Tr(L\e^{-tL})}{\Tr(\e^{-tL})}.
\DcalDefinition passage · 10 passages
Search aliases: Dcal D mathcal
Definition passage: Packet dephasing · Read passage · 02_strengthened.tex, line 992
[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.
Exact mathematical passage
\Dcal_{\Pcal}(X)=\sum_aP_aXP_a.
\Dcal_{\Pcal}Definition passage · 7 passages
Search aliases: Dcal_Pcal D_ P Dcal Pcal mathcal mathcal
Definition passage: Packet dephasing · Read passage · 02_strengthened.tex, line 992
[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.
Exact mathematical passage
\Dcal_{\Pcal}(X)=\sum_aP_aXP_a.
\ddDefinition passage · 30 passages
Search aliases: dd d mathrm
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1961
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
d_L(t) =-2\frac{\dd\log Z_L(t)}{\dd\log t} = 2t \frac{\Tr(L\e^{-tL})}{\Tr(\e^{-tL})}.
\dd^dDefinition passage · 1 passage
Search aliases: dd^d d^d dd mathrm
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
EDefinition passage · 129 passages
Search aliases: E
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\eDefinition passage · 48 passages
Search aliases: e mathrm
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1957
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
Z_L(t)=\Tr(\e^{-tL})
eDefinition passage · 65 passages
Search aliases: e
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\e^{-tL}Definition passage · 7 passages
Search aliases: e^-tL e mathrm
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1957
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
Z_L(t)=\Tr(\e^{-tL})
E_a^*Definition passage · 25 passages
Search aliases: E_a^*
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
E_bDefinition passage · 20 passages
Search aliases: E_b
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\EndDefinition passage · 19 passages
Search aliases: End operatorname
Definition passage: Finite quantum seed system · Read passage · 02_strengthened.tex, line 92
[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.
Exact mathematical passage
\Acal\subseteq\End(\Hcal)
FDefinition passage · 57 passages
Search aliases: F
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1506
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
F_P(z)=z-A-\Sigma(z).
fDefinition passage · 21 passages
Search aliases: f
Definition passage: Anomaly free continuum constraint system · Read passage · 02_strengthened.tex, line 2540
[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotient or physical subspace.
Exact mathematical passage
[C_a,C_b]\psi = \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
f_{ab}^{\phantom{ab}c}Definition passage · 6 passages
Search aliases: f_ab^phantomabc phantom
Definition passage: Anomaly free continuum constraint system · Read passage · 02_strengthened.tex, line 2540
[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotient or physical subspace.
Exact mathematical passage
[C_a,C_b]\psi = \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
F_PDefinition passage · 12 passages
Search aliases: F_P
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1506
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
F_P(z)=z-A-\Sigma(z).
GDefinition passage · 22 passages
Search aliases: G
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
gDefinition passage · 28 passages
Search aliases: g
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
HDefinition passage · 229 passages
Search aliases: H
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
H^*Definition passage · 9 passages
Search aliases: H^*
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 54
inition:001 A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
H=H^*
H_\ellDefinition passage · 8 passages
Search aliases: H_ℓ ell
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
H_\ell
\HcalDefinition passage · 62 passages
Search aliases: Hcal H mathcal
Definition passage: Finite quantum seed system · Read passage · 02_strengthened.tex, line 89
[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.
Exact mathematical passage
(\Hcal,\Acal,S),
\Hcal_\ellDefinition passage · 5 passages
Search aliases: Hcal_ℓ H_ℓ Hcal ell mathcal ell
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2344
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
\Hcal_\ell
\Hcal_{\ell+1}Definition passage · 3 passages
Search aliases: Hcal_ℓ+1 H_ℓ+1 Hcal ell mathcal ell
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2345
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
R_\ell:\Hcal_\ell\longrightarrow\Hcal_{\ell+1},
IDefinition passage · 187 passages
Search aliases: I
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
I_{\Ccal}Definition passage · 1 passage
Search aliases: I_Ccal I_ C Ccal mathcal
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
\iiDefinition passage · 69 passages
Search aliases: ii i mathrm
Definition passage: Anomaly free continuum constraint system · Read passage · 02_strengthened.tex, line 2540
[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotient or physical subspace.
Exact mathematical passage
[C_a,C_b]\psi = \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
\infDefinition passage · 2 passages
Search aliases: inf
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\inf_{c\in\CC}Definition passage · 2 passages
Search aliases: inf_c∈CC inf_c∈ C inf in CC inf in mathbb
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
KDefinition passage · 72 passages
Search aliases: K
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
kDefinition passage · 105 passages
Search aliases: k
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1285
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\ob_P(A) = PAP- \frac{\Tr(PAP)}{k}P.
LDefinition passage · 44 passages
Search aliases: L
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
lDefinition passage · 9 passages
Search aliases: l
Definition passage: Universal closure obstruction · Read passage · 02_strengthened.tex, line 2186
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
\Ob_{ab}^{\rm cl} = q_{\mathfrak c}([C_a,C_b]).
\logDefinition passage · 12 passages
Search aliases: log
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1961
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
d_L(t) =-2\frac{\dd\log Z_L(t)}{\dd\log t} = 2t \frac{\Tr(L\e^{-tL})}{\Tr(\e^{-tL})}.
\maxDefinition passage · 7 passages
Search aliases: max
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\max_{a,b}Definition passage · 1 passage
Search aliases: max_a,b max
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
NDefinition passage · 85 passages
Search aliases: N
Definition passage: Operator charge · Read passage · 02_strengthened.tex, line 294
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
q\in\ZZ/N\ZZ
\normDefinition passage · 45 passages
Search aliases: norm
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\ObDefinition passage · 4 passages
Search aliases: Ob operatorname
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1281
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\Ob_P(A)=q_P(PAP).
\obDefinition passage · 1 passage
Search aliases: ob o mathfrak
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1285
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\ob_P(A) = PAP- \frac{\Tr(PAP)}{k}P.
\Ob_{ab}^{\rm cl}Definition passage · 2 passages
Search aliases: Ob_ab^rm cl Ob rm operatorname rm
Definition passage: Universal closure obstruction · Read passage · 02_strengthened.tex, line 2186
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
\Ob_{ab}^{\rm cl} = q_{\mathfrak c}([C_a,C_b]).
\Ob_PDefinition passage · 2 passages
Search aliases: Ob_P Ob operatorname
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1281
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\Ob_P(A)=q_P(PAP).
\ob_PDefinition passage · 1 passage
Search aliases: ob_P o_P ob mathfrak
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1285
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\ob_P(A) = PAP- \frac{\Tr(PAP)}{k}P.
PDefinition passage · 512 passages
Search aliases: P
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
P_aDefinition passage · 97 passages
Search aliases: P_a
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 944
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
(P_a)
P_bDefinition passage · 47 passages
Search aliases: P_b
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 945
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Pcal_{ab}(X)=P_aXP_b.
P_{\ell,a}Definition passage · 3 passages
Search aliases: P_ℓ,a ell
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
P_{\ell,a}
P_{\ell,\alpha}Definition passage · 4 passages
Search aliases: P_ℓ,α ell alpha
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\PcalDefinition passage · 20 passages
Search aliases: Pcal P mathcal
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 945
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Pcal_{ab}(X)=P_aXP_b.
\Pcal_{ab}Definition passage · 4 passages
Search aliases: Pcal_ab P_ab Pcal mathcal
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 945
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Pcal_{ab}(X)=P_aXP_b.
\mathfrak QDefinition passage · 2 passages
Search aliases: Q mathfrak
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
qDefinition passage · 52 passages
Search aliases: q
Definition passage: Operator charge · Read passage · 02_strengthened.tex, line 294
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
q\in\ZZ/N\ZZ
q_{\mathfrak c}Definition passage · 2 passages
Search aliases: q_ c mathfrak
Definition passage: Universal closure obstruction · Read passage · 02_strengthened.tex, line 2182
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
q_{\mathfrak c}:\End(\Hcal)\longrightarrow\End(\Hcal)/\mathfrak c
q_PDefinition passage · 2 passages
Search aliases: q_P
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
RDefinition passage · 57 passages
Search aliases: R
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2345
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
R_\ell:\Hcal_\ell\longrightarrow\Hcal_{\ell+1},
rDefinition passage · 127 passages
Search aliases: r
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
C_{\ell,r}
R_\ellDefinition passage · 13 passages
Search aliases: R_ℓ ell
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2345
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
R_\ell:\Hcal_\ell\longrightarrow\Hcal_{\ell+1},
\rmDefinition passage · 39 passages
Search aliases: rm
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
SDefinition passage · 88 passages
Search aliases: S
Definition passage: Finite quantum seed system · Read passage · 02_strengthened.tex, line 89
[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.
Exact mathematical passage
(\Hcal,\Acal,S),
S^*Definition passage · 19 passages
Search aliases: S^*
Definition passage: Operator charge · Read passage · 02_strengthened.tex, line 295
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
SAS^*=\omega^qA.
\operatorname{span}Definition passage · 7 passages
Search aliases: span operatorname
Definition passage: Finite quantum seed system · Read passage · 02_strengthened.tex, line 93
[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.
Exact mathematical passage
\Acal S = \operatorname{span}\{A\psi:A\in\Acal,\ \psi\in S\}.
\SpecDefinition passage · 14 passages
Search aliases: Spec operatorname
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1501
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
z\notin\Spec D
TDefinition passage · 92 passages
Search aliases: T
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
T_\ell
tDefinition passage · 229 passages
Search aliases: t
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1957
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
Z_L(t)=\Tr(\e^{-tL})
T_\ellDefinition passage · 2 passages
Search aliases: T_ℓ ell
Definition passage: Finite quantum refinement tower · Read passage · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
T_\ell
\TrDefinition passage · 37 passages
Search aliases: Tr operatorname
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1285
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\ob_P(A) = PAP- \frac{\Tr(PAP)}{k}P.
XDefinition passage · 125 passages
Search aliases: X
Definition passage: Two sided operator packet · Read passage · 02_strengthened.tex, line 945
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Pcal_{ab}(X)=P_aXP_b.
xDefinition passage · 106 passages
Search aliases: x
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
ZDefinition passage · 41 passages
Search aliases: Z
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1957
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
Z_L(t)=\Tr(\e^{-tL})
zDefinition passage · 100 passages
Search aliases: z
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1501
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
z\notin\Spec D
Z_LDefinition passage · 4 passages
Search aliases: Z_L
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1957
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
Z_L(t)=\Tr(\e^{-tL})
\ZZDefinition passage · 11 passages
Search aliases: ZZ Z mathbb
Definition passage: Operator charge · Read passage · 02_strengthened.tex, line 294
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
q\in\ZZ/N\ZZ
\alphaDefinition passage · 25 passages
Search aliases: α alpha
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\GammaDefinition passage · 5 passages
Search aliases: Γ Gamma
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\Gamma_{\rm eff}Definition passage · 1 passage
Search aliases: Γ_rm eff Gamma rm
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\varepsilonDefinition passage · 33 passages
Search aliases: ε varepsilon
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\varepsilon_{\rm KL}Definition passage · 4 passages
Search aliases: ε_rm KL varepsilon rm
Definition passage: Approximate Knill Laflamme defect · Read passage · 02_strengthened.tex, line 1423
[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\LambdaDefinition passage · 5 passages
Search aliases: Λ Lambda
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\lambdaDefinition passage · 62 passages
Search aliases: λ lambda
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1956
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
\lambda_1,\ldots,\lambda_N
\lambda_1Definition passage · 2 passages
Search aliases: λ_1 lambda
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1956
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
\lambda_1,\ldots,\lambda_N
\lambda_NDefinition passage · 11 passages
Search aliases: λ_N lambda
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1956
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
\lambda_1,\ldots,\lambda_N
\piDefinition passage · 53 passages
Search aliases: π pi
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\SigmaDefinition passage · 20 passages
Search aliases: Σ Sigma
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1502
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
\Sigma(z)=B(z-D)^{-1}B^*
\tauDefinition passage · 24 passages
Search aliases: τ tau
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\psiDefinition passage · 107 passages
Search aliases: ψ psi
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\psi_0Definition passage · 15 passages
Search aliases: ψ_0 psi
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\omegaDefinition passage · 21 passages
Search aliases: ω omega
Definition passage: Operator charge · Read passage · 02_strengthened.tex, line 295
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
SAS^*=\omega^qA.
\omega^qDefinition passage · 3 passages
Search aliases: ω^q omega
Definition passage: Operator charge · Read passage · 02_strengthened.tex, line 295
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
SAS^*=\omega^qA.
\ldotsDefinition passage · 31 passages
Search aliases: … ldots
Definition passage: Heat trace and spectral dimension · Read passage · 02_strengthened.tex, line 1956
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
\lambda_1,\ldots,\lambda_N
\ellDefinition passage · 52 passages
Search aliases: ℓ ell
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\inDefinition passage · 89 passages
Search aliases: ∈ in
Definition passage: Finite quantum seed system · Read passage · 01_finite_core.tex, line 54
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\psi_0\in\cH
\notinDefinition passage · 3 passages
Search aliases: ∉ notin
Definition passage: Feshbach map and self energy · Read passage · 02_strengthened.tex, line 1501
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
z\notin\Spec D
\sumDefinition passage · 187 passages
Search aliases: ∑ sum
Definition passage: Packet dephasing · Read passage · 02_strengthened.tex, line 992
[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.
Exact mathematical passage
\Dcal_{\Pcal}(X)=\sum_aP_aXP_a.
\sum_aDefinition passage · 63 passages
Search aliases: ∑_a sum
Definition passage: Packet dephasing · Read passage · 02_strengthened.tex, line 992
[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.
Exact mathematical passage
\Dcal_{\Pcal}(X)=\sum_aP_aXP_a.
\sum_cDefinition passage · 8 passages
Search aliases: ∑_c sum
Definition passage: Anomaly free continuum constraint system · Read passage · 02_strengthened.tex, line 2540
[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotient or physical subspace.
Exact mathematical passage
[C_a,C_b]\psi = \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
\intDefinition passage · 14 passages
Search aliases: ∫ int
Definition passage: Einstein infrared criterion · Read passage · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\subseteqDefinition passage · 6 passages
Search aliases: ⊆ subseteq
Definition passage: Finite quantum seed system · Read passage · 02_strengthened.tex, line 92
[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.
Exact mathematical passage
\Acal\subseteq\End(\Hcal)
\longrightarrowDefinition passage · 23 passages
Search aliases: ⟶ longrightarrow
Definition passage: Code compression obstruction · Read passage · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
\mathcal AIntroduction passage · 5 passages
Search aliases: A mathcal
Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
A^*Introduction passage · 4 passages
Search aliases: A^*
Introduction passage · monograph_atlas.tex, line 20
vpage2The objects on the pageAN ATLAS OF TYPES AND CONVENTIONSTraditional edition: Definition II.2.1, Definition II.7.1. The same letter can stand for different kinds of data in different calculations. Keep the following types fixed while reading a formula. In this appendix, vectors are columns, operators compose from right to left, and A^* denotes the conjugate transpose. linewidth@p.13linewidthYY@ Symbol · Object · What it records
Exact mathematical passage
A^*
A_+Introduction passage · 4 passages
Search aliases: A_+
Introduction passage · Read passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
Exact mathematical passage
Q=I-\sum_{r\in A_+}V_rV_r^*
a_1Introduction passage · 11 passages
Search aliases: a_1
Introduction passage · Read passage · 01_finite_core.tex, line 161
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
Exact mathematical passage
\left\{ X_1^{a_1}\cdots X_n^{a_n}\psi_0: (a_1,\ldots,a_n)\in\{0,1\}^n \right\}
A_aIntroduction passage · 5 passages
Search aliases: A_a
Introduction passage · Read passage · 03_supplement_06.tex, line 38
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
Exact mathematical passage
A_a(x):\mathcal E_x\longrightarrow\mathcal E_a, \quad w\longmapsto P(a)T_a(x)^{-1}w.
a_nIntroduction passage · 5 passages
Search aliases: a_n
Introduction passage · Read passage · 01_finite_core.tex, line 161
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
Exact mathematical passage
\left\{ X_1^{a_1}\cdots X_n^{a_n}\psi_0: (a_1,\ldots,a_n)\in\{0,1\}^n \right\}
B_aIntroduction passage · 3 passages
Search aliases: B_a
Introduction passage · Read passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
Exact mathematical passage
B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
\CIntroduction passage · 10 passages
Search aliases: C mathbb
Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
C^{\mathsf T}Introduction passage · 1 passage
Search aliases: C^ T mathsf
Introduction passage · Read passage · 03_supplement_03.tex, line 74
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
Exact mathematical passage
\rho_B=C^{\mathsf T}(C^{\mathsf T})^*, \qquad \rank\rho_B=\rank C=r(\psi).
C^*Introduction passage · 5 passages
Search aliases: C^*
Introduction passage · Read passage · 03_supplement_03.tex, line 78
al trace and its support The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
Exact mathematical passage
C^*C
\CC^2Introduction passage · 10 passages
Search aliases: CC^2 C^2 CC mathbb
Introduction passage · monograph_atlas.tex, line 712
al values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every finite eigenvalue of H' exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space …
Exact mathematical passage
\CC^2
\operatorname{CNOT}Introduction passage · 1 passage
Search aliases: CNOT operatorname
Introduction passage · Read passage · 01_finite_core.tex, line 337
[Bell seed circuit] In this example only, H denotes the Hadamard gate, not a Hamiltonian. Starting from ket00, (H⊗ I)ket00 = fracket00+ket10sqrt2, and then CNOT(H⊗ I)ket00 = fracket00+ket11sqrt2. The parity packets show that the final state lies entirely in the even sector, while the intermediate circuit path passes through both even and odd addresses.
Exact mathematical passage
\operatorname{CNOT}(H\otimes I)\ket{00} = \frac{\ket{00}+\ket{11}}{\sqrt2}.
D_{[X,Y]}Introduction passage · 1 passage
Search aliases: D_[X,Y]
Introduction passage · Read passage · 01_finite_core.tex, line 617
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
Exact mathematical passage
D_XD_Ys-D_YD_Xs-D_{[X,Y]}s=0.
d_iIntroduction passage · 16 passages
Search aliases: d_i
Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
D_MIntroduction passage · 4 passages
Search aliases: D_M
Introduction passage · Read passage · 03_supplement_06.tex, line 61
A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.
Exact mathematical passage
\nabla_Xs(x)=P(x)\,D_Ms(x)[X(x)].
D_XIntroduction passage · 3 passages
Search aliases: D_X
Introduction passage · Read passage · 01_finite_core.tex, line 616
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
Exact mathematical passage
D_Xs=ds(X)
D_YIntroduction passage · 3 passages
Search aliases: D_Y
Introduction passage · Read passage · 01_finite_core.tex, line 617
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
Exact mathematical passage
D_XD_Ys-D_YD_Xs-D_{[X,Y]}s=0.
d_\lambdaIntroduction passage · 6 passages
Search aliases: d_λ lambda
Introduction passage · Read passage · 02_strengthened.tex, line 203
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
Exact mathematical passage
d_\lambda
\diagIntroduction passage · 27 passages
Search aliases: diag operatorname
Introduction passage · monograph_atlas.tex, line 705
ermines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_…
Exact mathematical passage
T^n=\diag(1,2^{-n},4^{-n}),\qquad P_0=\diag(1,0,0),\qquad\norm{T^n-P_0}=2^{-n}.
\dot UIntroduction passage · 16 passages
Search aliases: dot U dot
Introduction passage · Read passage · 03_supplement_02.tex, line 90
Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.
Exact mathematical passage
\dot U
\dot\psiIntroduction passage · 4 passages
Search aliases: dotψ dot psi
Introduction passage · Read passage · 03_supplement_02.tex, line 90
Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.
Exact mathematical passage
\dot\psi
\mathcal EIntroduction passage · 9 passages
Search aliases: E mathcal
Introduction passage · Read passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
Exact mathematical passage
B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
e^{-H}Introduction passage · 1 passage
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Introduction passage · monograph_atlas.tex, line 704
vpage36A positive transfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly …
Exact mathematical passage
T=e^{-H}
e^{-H'}Introduction passage · 1 passage
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Introduction passage · monograph_atlas.tex, line 712
integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every finite eigenvalue of H' exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the l…
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T'=e^{-H'}
e^{-i(t-s)H}Introduction passage · 1 passage
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Introduction passage · Read passage · 04_alternative_derivations.tex, line 12
ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior…
Exact mathematical passage
[e^{-itH},P]=-i\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\,ds.
e^{-isH}Introduction passage · 3 passages
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Introduction passage · Read passage · 04_alternative_derivations.tex, line 12
ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior…
Exact mathematical passage
[e^{-itH},P]=-i\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\,ds.
e^{-itH}Introduction passage · 7 passages
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Introduction passage · Read passage · 04_alternative_derivations.tex, line 12
ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior…
Exact mathematical passage
[e^{-itH},P]=-i\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\,ds.
e^{-n\Delta}Introduction passage · 1 passage
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Introduction passage · monograph_atlas.tex, line 709
HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, b…
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e^{-n\Delta}
E_aIntroduction passage · 36 passages
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Introduction passage · Read passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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(E_a)
\mathcal E_aIntroduction passage · 5 passages
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Introduction passage · Read passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
Exact mathematical passage
B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
\mathcal E_xIntroduction passage · 5 passages
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Introduction passage · Read passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
Exact mathematical passage
B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
\mathcal HIntroduction passage · 3 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
H'Introduction passage · 3 passages
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Introduction passage · monograph_atlas.tex, line 712
positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every finite eigenvalue of H' exponentiates to a strictly positive value. One may restrict to the positive support, but that cha…
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H'
iIntroduction passage · 92 passages
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Introduction passage · Read passage · 03_supplement_02.tex, line 91
Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.
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i\dot\psi=H\psi(t).
I_AIntroduction passage · 2 passages
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Introduction passage · Read passage · 03_supplement_05.tex, line 146
An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R widetilde N_E(ρ)=ρ. This includes entangled density matrices s…
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\widetilde{\mathcal N}_E(\rho) =\sum_a(I_A\otimes E_a)\rho(I_A\otimes E_a)^*.
I_{m_i}Introduction passage · 3 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
jIntroduction passage · 138 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 160
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
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X_j
\ketIntroduction passage · 99 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 157
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
Exact mathematical passage
\psi_0=\ket{0}^{\otimes n}
\lceilIntroduction passage · 9 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
MIntroduction passage · 33 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
mIntroduction passage · 98 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
M_{d_i}Introduction passage · 3 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
m_iIntroduction passage · 16 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
\max_iIntroduction passage · 4 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
\McalIntroduction passage · 4 passages
Search aliases: Mcal M mathcal
Introduction passage · Read passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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\Mcal\subseteq P\End(\Hcal)P
\Mcal'Introduction passage · 1 passage
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Introduction passage · Read passage · 02_strengthened.tex, line 1485
hannel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H)P [source reference].
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\Mcal'
nIntroduction passage · 119 passages
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Introduction passage · monograph_atlas.tex, line 704
ransfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xf…
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n\geq0
\overline{\psi_{i\ell}}Introduction passage · 1 passage
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Introduction passage · Read passage · 03_supplement_03.tex, line 70
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
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(\rho_B)_{j\ell}=\sum_i\psi_{ij}\overline{\psi_{i\ell}}.
P^2Introduction passage · 10 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 616
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
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P^2=P
P_0Introduction passage · 9 passages
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Introduction passage · monograph_atlas.tex, line 705
ermines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_…
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T^n=\diag(1,2^{-n},4^{-n}),\qquad P_0=\diag(1,0,0),\qquad\norm{T^n-P_0}=2^{-n}.
P_\alphaIntroduction passage · 16 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
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P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
P_\alpha^*Introduction passage · 1 passage
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Introduction passage · Read passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
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P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
P_\betaIntroduction passage · 7 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
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P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
QIntroduction passage · 97 passages
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Introduction passage · Read passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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Q=I-\sum_{r\in A_+}V_rV_r^*
Q^*Introduction passage · 2 passages
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Introduction passage · Read passage · 03_supplement_05.tex, line 76
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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\sum_{r\in A_+}V_rV_r^*+Q^*Q=I.
\mathcal RIntroduction passage · 14 passages
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Introduction passage · Read passage · 03_supplement_05.tex, line 72
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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\boxed{\mathcal R(\sigma)=\sum_{r\in A_+}V_r^*\sigma V_r+Q\sigma Q.}
\rankIntroduction passage · 17 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 74
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
Exact mathematical passage
\rho_B=C^{\mathsf T}(C^{\mathsf T})^*, \qquad \rank\rho_B=\rank C=r(\psi).
\RcalIntroduction passage · 11 passages
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Introduction passage · Read passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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\Rcal
\Rcal^*Introduction passage · 1 passage
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Introduction passage · Read passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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P(\Phi^*\circ\Rcal^*)(X)P=X
\rceilIntroduction passage · 9 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
sIntroduction passage · 96 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 616
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
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D_Xs=ds(X)
\srankIntroduction passage · 1 passage
Search aliases: srank operatorname
Introduction passage · Read passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
\srank_{\mathcal A}Introduction passage · 1 passage
Search aliases: srank_ A srank mathcal operatorname mathcal
Introduction passage · Read passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
\mathsf TIntroduction passage · 14 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 74
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
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\rho_B=C^{\mathsf T}(C^{\mathsf T})^*, \qquad \rank\rho_B=\rank C=r(\psi).
T'Introduction passage · 2 passages
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Introduction passage · monograph_atlas.tex, line 712
lidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every finite eigenvalue of H' exponentia…
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T'=\diag(1,0)
T^nIntroduction passage · 6 passages
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Introduction passage · monograph_atlas.tex, line 705
ermines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_…
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T^n=\diag(1,2^{-n},4^{-n}),\qquad P_0=\diag(1,0,0),\qquad\norm{T^n-P_0}=2^{-n}.
T_aIntroduction passage · 15 passages
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Introduction passage · Read passage · 03_supplement_06.tex, line 33
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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T_a(x)=T(P(x),P(a))
T_xIntroduction passage · 1 passage
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Introduction passage · Read passage · 03_supplement_06.tex, line 65
A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.
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\nabla s(x):T_xM\longrightarrow\mathcal E_x.
UIntroduction passage · 104 passages
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Introduction passage · Read passage · 02_strengthened.tex, line 200
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
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U:G\longrightarrow\Ucal(\Hcal)
U_aIntroduction passage · 7 passages
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Introduction passage · Read passage · 03_supplement_06.tex, line 33
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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x\in U_a
\UcalIntroduction passage · 7 passages
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Introduction passage · Read passage · 02_strengthened.tex, line 200
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
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U:G\longrightarrow\Ucal(\Hcal)
VIntroduction passage · 60 passages
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Introduction passage · Read passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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Q=I-\sum_{r\in A_+}V_rV_r^*
vIntroduction passage · 51 passages
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Introduction passage · Read passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
V_rIntroduction passage · 8 passages
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Introduction passage · Read passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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Q=I-\sum_{r\in A_+}V_rV_r^*
V_r^*Introduction passage · 7 passages
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Introduction passage · Read passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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Q=I-\sum_{r\in A_+}V_rV_r^*
wIntroduction passage · 18 passages
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Introduction passage · Read passage · 03_supplement_06.tex, line 38
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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A_a(x):\mathcal E_x\longrightarrow\mathcal E_a, \quad w\longmapsto P(a)T_a(x)^{-1}w.
\wedgeIntroduction passage · 10 passages
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Introduction passage · Read passage · 04_alternative_derivations.tex, line 22
eld and extend the projected connection to differential forms with values in the range bundle by ∇ω=P dω. Thus ∇^2 denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields. [Alternative proof of the curvature formula] For a section s with Ps=s, ∇^2s=P d(P ds)=P(dP)wedge ds. Since ds=d(Ps)=(dP)s+Pds and P(dP)P=0, the surviving term is P(dPwedge dP)Ps. Restricted low-mode proof. Let h≠0, x=kh/2, and 0≤ x≤1. [Alternative proof …
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\nabla^2s=P\,d(P\,ds)=P(dP)\wedge ds.
\widehat GIntroduction passage · 4 passages
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Introduction passage · Read passage · 02_strengthened.tex, line 203
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
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\widehat G
\widetilde{\mathcal N}Introduction passage · 2 passages
Search aliases: widetilde N widetilde mathcal
Introduction passage · Read passage · 03_supplement_05.tex, line 146
An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R widetilde N_E(ρ)=ρ. This includes entangled density matrices s…
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\widetilde{\mathcal N}_E(\rho) =\sum_a(I_A\otimes E_a)\rho(I_A\otimes E_a)^*.
\widetilde{\mathcal N}_EIntroduction passage · 2 passages
Search aliases: widetilde N_E widetilde mathcal
Introduction passage · Read passage · 03_supplement_05.tex, line 146
An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R widetilde N_E(ρ)=ρ. This includes entangled density matrices s…
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\widetilde{\mathcal N}_E(\rho) =\sum_a(I_A\otimes E_a)\rho(I_A\otimes E_a)^*.
\widetilde{\mathcal R}Introduction passage · 1 passage
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Introduction passage · Read passage · 03_supplement_05.tex, line 155
channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R widetilde N_E(ρ)=ρ. This includes entangled density matrices supported on the combined reference and code space. In fact, the identity holds for every j…
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\boxed{\widetilde{\mathcal R}\,\widetilde{\mathcal N}_E(\rho)=\rho.}
X_1^{a_1}Introduction passage · 4 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 161
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
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\left\{ X_1^{a_1}\cdots X_n^{a_n}\psi_0: (a_1,\ldots,a_n)\in\{0,1\}^n \right\}
X_jIntroduction passage · 6 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 160
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
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X_j
X_n^{a_n}Introduction passage · 3 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 161
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
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\left\{ X_1^{a_1}\cdots X_n^{a_n}\psi_0: (a_1,\ldots,a_n)\in\{0,1\}^n \right\}
YIntroduction passage · 23 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 617
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
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D_XD_Ys-D_YD_Xs-D_{[X,Y]}s=0.
|Introduction passage · 34 passages
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Introduction passage · Read passage · 04_alternative_derivations.tex, line 15
lex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior forms. Let P be a smooth idempotent field and extend the projected connection to differential forms with values in the …
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|t|
\betaIntroduction passage · 10 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
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P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
\DeltaIntroduction passage · 7 passages
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Introduction passage · monograph_atlas.tex, line 709
I.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the oper…
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\Delta=\log2
\deltaIntroduction passage · 43 passages
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Introduction passage · Read passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
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P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
\delta_{\alpha\beta}Introduction passage · 1 passage
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Introduction passage · Read passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
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P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
\rhoIntroduction passage · 90 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 70
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
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(\rho_B)_{j\ell}=\sum_i\psi_{ij}\overline{\psi_{i\ell}}.
\rho_BIntroduction passage · 5 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 70
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
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(\rho_B)_{j\ell}=\sum_i\psi_{ij}\overline{\psi_{i\ell}}.
\sigmaIntroduction passage · 7 passages
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Introduction passage · Read passage · 03_supplement_05.tex, line 72
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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\boxed{\mathcal R(\sigma)=\sum_{r\in A_+}V_r^*\sigma V_r+Q\sigma Q.}
\PhiIntroduction passage · 43 passages
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Introduction passage · Read passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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\Phi
\Phi^*Introduction passage · 1 passage
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Introduction passage · Read passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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P(\Phi^*\circ\Rcal^*)(X)P=X
\chiIntroduction passage · 7 passages
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Introduction passage · Read passage · 02_strengthened.tex, line 203
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
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\chi_\lambda
\chi_\lambdaIntroduction passage · 2 passages
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Introduction passage · Read passage · 02_strengthened.tex, line 203
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
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\chi_\lambda
\psi_{ij}Introduction passage · 3 passages
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Introduction passage · Read passage · 03_supplement_03.tex, line 70
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
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(\rho_B)_{j\ell}=\sum_i\psi_{ij}\overline{\psi_{i\ell}}.
\hbarIntroduction passage · 1 passage
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Introduction passage · Read passage · 01_finite_core.tex, line 10
of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real derivatives. A matrix X is supported on a code projector P when PXP=X.
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\hbar=1
\nablaIntroduction passage · 20 passages
Search aliases: ∇ nabla
Introduction passage · Read passage · 03_supplement_06.tex, line 61
A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.
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\nabla_Xs(x)=P(x)\,D_Ms(x)[X(x)].
\nabla^2Introduction passage · 2 passages
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Introduction passage · Read passage · 04_alternative_derivations.tex, line 19
rms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior forms. Let P be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by ∇ω=P dω. Thus ∇^2 denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields. [Alternative proof of the curvature formula] For a section s with Ps=s, ∇^2s=P d(P ds)=P(dP)…
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\nabla^2
\nabla_XIntroduction passage · 11 passages
Search aliases: ∇_X nabla
Introduction passage · Read passage · 03_supplement_06.tex, line 61
A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.
Exact mathematical passage
\nabla_Xs(x)=P(x)\,D_Ms(x)[X(x)].
\sum_iIntroduction passage · 3 passages
Search aliases: ∑_i sum
Introduction passage · Read passage · 03_supplement_03.tex, line 70
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
Exact mathematical passage
(\rho_B)_{j\ell}=\sum_i\psi_{ij}\overline{\psi_{i\ell}}.
\sum_{r\in A_+}Introduction passage · 3 passages
Search aliases: ∑_r∈ A_+ sum in
Introduction passage · Read passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
Exact mathematical passage
Q=I-\sum_{r\in A_+}V_rV_r^*
\sum_\alphaIntroduction passage · 8 passages
Search aliases: ∑_α sum alpha
Introduction passage · Read passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
Exact mathematical passage
P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
\circIntroduction passage · 3 passages
Search aliases: ∘ circ
Introduction passage · Read passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
Exact mathematical passage
P(\Phi^*\circ\Rcal^*)(X)P=X
\int_0^tIntroduction passage · 10 passages
Search aliases: ∫_0^t int
Introduction passage · Read passage · 04_alternative_derivations.tex, line 12
ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior…
Exact mathematical passage
[e^{-itH},P]=-i\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\,ds.
\leIntroduction passage · 39 passages
Search aliases: ≤ le
Introduction passage · Read passage · 03_supplement_03.tex, line 34
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
m_i\le kd_i
\geIntroduction passage · 33 passages
Search aliases: ≥ ge
Introduction passage · Read passage · 01_finite_core.tex, line 156
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
Exact mathematical passage
n\ge0
\geqIntroduction passage · 17 passages
Search aliases: ≥ geq
Introduction passage · monograph_atlas.tex, line 704
ransfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xf…
Exact mathematical passage
n\geq0
\bigoplusIntroduction passage · 8 passages
Search aliases: ⊕ bigoplus
Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
\bigoplus_iIntroduction passage · 2 passages
Search aliases: ⊕_i bigoplus
Introduction passage · Read passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
\otimesIntroduction passage · 74 passages
Search aliases: ⊗ otimes
Introduction passage · Read passage · 01_finite_core.tex, line 157
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
Exact mathematical passage
\psi_0=\ket{0}^{\otimes n}
\cdotsIntroduction passage · 21 passages
Search aliases: ⋯ cdots
Introduction passage · Read passage · 01_finite_core.tex, line 161
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
Exact mathematical passage
\left\{ X_1^{a_1}\cdots X_n^{a_n}\psi_0: (a_1,\ldots,a_n)\in\{0,1\}^n \right\}
\longmapstoIntroduction passage · 6 passages
Search aliases: ⟼ longmapsto
Introduction passage · Read passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
Exact mathematical passage
B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
\bm aUse in source · 1 passage
Search aliases: a bm
Use in source · Read passage · 02_strengthened.tex, line 920
The Pauli identity ( a·σ)( b·σ) =( a· b)I + i( a× b)·σ gives - iTr(P[∂_θ P,∂_ϕ P]) =frac12 n· (∂_θ n×∂_ϕ n) =frac12sinθ. Integrating the displayed curvature over the sphere and applying the stated normalization gives one.
Exact mathematical passage
(\bm a\cdot\bm\sigma)(\bm b\cdot\bm\sigma) =(\bm a\cdot\bm b)I +\ii(\bm a\times\bm b)\cdot\bm\sigma
\mathbf aUse in source · 1 passage
Search aliases: a mathbf
Use in source · Read passage · 01_finite_core.tex, line 651
P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.
Exact mathematical passage
\Tr(P\,\mathbf a\cdot\boldsymbol\sigma)=\mathbf n\cdot\mathbf a
\mathfrak AUse in source · 7 passages
Search aliases: A mathfrak
Use in source · Read passage · 01_finite_core.tex, line 1199
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\boxed{ \mathfrak A_{AB} = [\widehat C_A,\widehat C_B] -i\sum_C f_{AB}^{\ \ C}\widehat C_C. }
\mathfrak A^{(\ell)}_{AB}Use in source · 1 passage
Search aliases: A^(ℓ)_AB mathfrak ell
Use in source · Read passage · 01_finite_core.tex, line 1223
[Anomaly-free refinement] For a specified nontrivial quantum-geometric constraint model, there exists a coherent packet tower with nonzero physical spaces, compatible physical projectors and exact operator closure A^(ℓ)_AB=0 at every finite level. The compatible physical compression then vanishes as a consequence, not as a substitute for closure. The substantive research target is the operator condition. The earlier condition stated only in terms of compression was too weak. A choice of model, d…
Exact mathematical passage
\mathfrak A^{(\ell)}_{AB}=0
a^{-1}Use in source · 1 passage
Search aliases: a^-1
Use in source · Read passage · 02_strengthened.tex, line 1860
[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.
Exact mathematical passage
H_T=-a^{-1}\log T
A^2Use in source · 3 passages
Search aliases: A^2
Use in source · monograph_atlas.tex, line 694
ct worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i), so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors. headWhy this does not close to a bare local equation With hidden initial state zero, the full system has ddot x(0)=-(A^2+BB^*)x(0)=-x(0). The bare retained equation idot x=Ax would instead give zero acceleration. Therefor…
Exact mathematical passage
\ddot x(0)=-(A^2+BB^*)x(0)=-x(0)
a_0Use in source · 10 passages
Search aliases: a_0
Use in source · Read passage · 02_strengthened.tex, line 389
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
a_0
A_1Use in source · 3 passages
Search aliases: A_1
Use in source · Read passage · 02_strengthened.tex, line 366
[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.
Exact mathematical passage
A_1\otimes\cdots\otimes A_m
a_2Use in source · 2 passages
Search aliases: a_2
Use in source · monograph_atlas.tex, line 44
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'…
Exact mathematical passage
a=(a_1,a_2,a_3)
a_3Use in source · 2 passages
Search aliases: a_3
Use in source · monograph_atlas.tex, line 44
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'…
Exact mathematical passage
a=(a_1,a_2,a_3)
\mathcal A_{ab}Use in source · 3 passages
Search aliases: A_ab mathcal
Use in source · Read passage · 03_supplement_01.tex, line 189
A constraint diagnostic that vanishes automatically. Section 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives C_aP=0 for every a. Absorb the factor i and any structure coefficient into f_c, and write A_ab=C_aC_b-C_bC_a-∑_cf_cC_c. [Exact common-kernel compression] If C_cP=0 for every c, then A_abP=0, P A_abP=0. No idempotence or adjoint property of P is needed for this implication. Associativity gives A_abP=C_a(C_bP)-C_b(C_aP)-∑_c…
Exact mathematical passage
\mathcal A_{ab}=C_aC_b-C_bC_a-\sum_cf_cC_c.
\mathfrak A_{AB}Use in source · 3 passages
Search aliases: A_AB mathfrak
Use in source · Read passage · 01_finite_core.tex, line 1199
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\boxed{ \mathfrak A_{AB} = [\widehat C_A,\widehat C_B] -i\sum_C f_{AB}^{\ \ C}\widehat C_C. }
\mathfrak A_{ab}Use in source · 2 passages
Search aliases: A_ab mathfrak
Use in source · Read passage · 02_strengthened.tex, line 2206
If structure coefficients f_ab^phantomabc are prescribed, define the fixed structure anomaly A_ab = [C_a,C_b] - i∑_cf_ab^phantomabcC_c. Exact realization of the prescribed algebra is the stronger condition A_ab=0.
Exact mathematical passage
\mathfrak A_{ab} = [C_a,C_b] -\ii\sum_cf_{ab}^{\phantom{ab}c}C_c.
A_bUse in source · 1 passage
Search aliases: A_b
Use in source · Read passage · 03_supplement_06.tex, line 49
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{ab}(x)=A_b(x)B_a(x)
A_{ba}Use in source · 1 passage
Search aliases: A_ba
Use in source · Read passage · 02_strengthened.tex, line 12
The elementary coordinate object is a complete family of orthogonal projectors P_aP_b=δ_abP_a, ∑_a∈ IP_a=I. Every vector has packet coordinates P_aψ. Every operator has transfer blocks A_ba=P_bAP_a. These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction, effective dynamics, and geometric limits.
Exact mathematical passage
A_{ba}=P_bAP_a.
A_iUse in source · 3 passages
Search aliases: A_i
Use in source · Read passage · 03_supplement_02.tex, line 47
Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite number of factors with the stated finite coordinate sets.
Exact mathematical passage
A_i
a_iUse in source · 1 passage
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Use in source · Read passage · 03_supplement_02.tex, line 54
Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.
Exact mathematical passage
Q_a=\mathsf T\bigl(i\mapsto P^{(i)}_{a_i}\bigr), \qquad C_r=\sum_{\sum_i a_i=r}Q_a.
A_jUse in source · 2 passages
Search aliases: A_j
Use in source · Read passage · 02_strengthened.tex, line 1903
[Reflection positive correlators] Fix a vector Ω and operators A_1,…,A_m. The matrix C_jk = ipA_jΩT^n_j+n_kA_kΩ is positive semidefinite.
Exact mathematical passage
C_{jk} = \ip{A_j\Omega}{T^{n_j+n_k}A_k\Omega}
a_jUse in source · 5 passages
Search aliases: a_j
Use in source · Read passage · 03_supplement_02.tex, line 17
0 · x≠ a. The seed is ψ_0=ket0. Define a translation operator by (T_a v)(x)=v(x-a). At position j, the local Pauli X is X_j=T_e_j, where e_j has a single nonzero binary coordinate. The formal circuit uses the identity when a_j=0 and X_j when a_j=1.
Exact mathematical passage
a_j=0
A_kUse in source · 2 passages
Search aliases: A_k
Use in source · Read passage · 02_strengthened.tex, line 1903
[Reflection positive correlators] Fix a vector Ω and operators A_1,…,A_m. The matrix C_jk = ipA_jΩT^n_j+n_kA_kΩ is positive semidefinite.
Exact mathematical passage
C_{jk} = \ip{A_j\Omega}{T^{n_j+n_k}A_k\Omega}
A_mUse in source · 2 passages
Search aliases: A_m
Use in source · Read passage · 02_strengthened.tex, line 366
[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.
Exact mathematical passage
A_1\otimes\cdots\otimes A_m
a_mUse in source · 7 passages
Search aliases: a_m
Use in source · Read passage · 02_strengthened.tex, line 389
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
a_m
a_{m-1}Use in source · 3 passages
Search aliases: a_m-1
Use in source · Read passage · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
A_NUse in source · 1 passage
Search aliases: A_N
Use in source · monograph_atlas.tex, line 284
ndomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ≤θ_0 is π(1-cosθ_0). At θ_0=π/3, it is exactly π/2. The latitude drawn in the plate il…
Exact mathematical passage
A_N=-\ii u_N^*\dd u_N=\frac{1-\cos\theta}{2}\,\dd\phi, \qquad \dd A_N=\frac12\sin\theta\,\dd\theta\wedge\dd\phi.
A_nUse in source · 10 passages
Search aliases: A_n
Use in source · monograph_atlas.tex, line 795
one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how finite quadratic forms pass positivity to the limit. xfigtrace_topology headOperator norm does not preserve trace in growing rank On ℓ^2, let P_n select the first n coordinates and set A_n=P_n/n. Then normA_n=1/n→0 while Tr A_n=1 for every n. These positive trace-class operators show why the choice of topology matters. Th…
Exact mathematical passage
A_n=P_n/n
A_{ra}Use in source · 3 passages
Search aliases: A_ra
Use in source · Read passage · 03_supplement_05.tex, line 87
Deriving the Gram matrix from recovery. Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code implies ∑_r,a A_raρ A_ra^*=Pρ P. This equality on all matrices follows by applying correction to the sandwich supported on the code, Pρ P, using the …
Exact mathematical passage
A_{ra}=R_rE_aP.
A_{ra}^*Use in source · 1 passage
Search aliases: A_ra^*
Use in source · Read passage · 03_supplement_05.tex, line 91
Deriving the Gram matrix from recovery Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code implies ∑_r,a A_raρ A_ra^*=Pρ P. This equality on all matrices follows by applying correction to the sandwich supported on the code, Pρ P, using the p…
Exact mathematical passage
\sum_{r,a} A_{ra}\rho A_{ra}^*=P\rho P.
A_wUse in source · 4 passages
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Use in source · Read passage · 01_finite_core.tex, line 802
We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix A_w.
Exact mathematical passage
A_w=\sum_aw_aE_a
A_w^*Use in source · 2 passages
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Use in source · Read passage · 01_finite_core.tex, line 803
We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix A_w.
Exact mathematical passage
PXP=X\quad\Longrightarrow\quad \cR(A_wXA_w^*)=t_wX.
\mathfrak A_{\ell+1,ab}Use in source · 1 passage
Search aliases: A_ℓ+1,ab mathfrak ell
Use in source · Read passage · 02_strengthened.tex, line 2279
[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.
Exact mathematical passage
\mathfrak A_{\ell+1,ab}R_\ell = R_\ell\mathfrak A_{\ell,ab}.
\mathfrak A_{\ell,ab}Use in source · 1 passage
Search aliases: A_ℓ,ab mathfrak ell
Use in source · Read passage · 02_strengthened.tex, line 2279
[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.
Exact mathematical passage
\mathfrak A_{\ell+1,ab}R_\ell = R_\ell\mathfrak A_{\ell,ab}.
\AAUse in source · 2 passages
Search aliases: AA
Use in source · monograph_atlas.tex, line 26
AA · Declared operator algebra · It specifies the available linear operations.
Exact mathematical passage
\AA
\Acal'Use in source · 1 passage
Search aliases: Acal' A' Acal mathcal
Use in source · Read passage · 02_strengthened.tex, line 863
[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.
Exact mathematical passage
\Acal' = G^*\Acal G+G^*\dd G.
\Acal^*Use in source · 1 passage
Search aliases: Acal^* A^* Acal mathcal
Use in source · Read passage · 02_strengthened.tex, line 879
Differentiate V^*V=I to obtain A^*=- A. The gauge transformation law follows by differentiating VG. Since P=VV^* and (I-P) d V= d P V, direct expansion gives d A+ Awedge A = d V^*(I-P)wedge d V = V^*( d Pwedge d P)V.
Exact mathematical passage
\Acal^*=-\Acal
\Acal_{\Pcal}Use in source · 1 passage
Search aliases: Acal_Pcal A_ P Acal Pcal mathcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 999
[Dephasing as conditional expectation] The map D_ P is completely positive, trace preserving, unital, and idempotent. It is the Hilbert Schmidt orthogonal projection onto the block diagonal algebra A_ P = {X:[X,P_a]=0 for every a}. In particular, norm D_ P(X)_HS ≤ normX_HS.
Exact mathematical passage
\Acal_{\Pcal} = \{X:[X,P_a]=0\text{ for every }a\}.
\AdUse in source · 4 passages
Search aliases: Ad operatorname
Use in source · Read passage · 02_strengthened.tex, line 1093
Let S be unitary and N≥1 an integer with S^N=I, and define the conjugation action Ad_S(X)=SXS^*.
Exact mathematical passage
\Ad_S(X)=SXS^*.
\Ad_SUse in source · 3 passages
Search aliases: Ad_S Ad operatorname
Use in source · Read passage · 02_strengthened.tex, line 1093
Let S be unitary and N≥1 an integer with S^N=I, and define the conjugation action Ad_S(X)=SXS^*.
Exact mathematical passage
\Ad_S(X)=SXS^*.
\Ad_{U(g)}Use in source · 1 passage
Search aliases: Ad_U(g) Ad operatorname
Use in source · Read passage · 02_strengthened.tex, line 1166
Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.
Exact mathematical passage
\Ad_{U(g)}\circ\Phi\circ\Ad_{U(g)^*}.
\Ad_{U(g)^*}Use in source · 1 passage
Search aliases: Ad_U(g)^* Ad operatorname
Use in source · Read passage · 02_strengthened.tex, line 1166
Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.
Exact mathematical passage
\Ad_{U(g)}\circ\Phi\circ\Ad_{U(g)^*}.
\bm bUse in source · 1 passage
Search aliases: b bm
Use in source · Read passage · 02_strengthened.tex, line 920
The Pauli identity ( a·σ)( b·σ) =( a· b)I + i( a× b)·σ gives - iTr(P[∂_θ P,∂_ϕ P]) =frac12 n· (∂_θ n×∂_ϕ n) =frac12sinθ. Integrating the displayed curvature over the sphere and applying the stated normalization gives one.
Exact mathematical passage
(\bm a\cdot\bm\sigma)(\bm b\cdot\bm\sigma) =(\bm a\cdot\bm b)I +\ii(\bm a\times\bm b)\cdot\bm\sigma
B^2Use in source · 1 passage
Search aliases: B^2
Use in source · Read passage · 02_strengthened.tex, line 1640
[Low energy expansion] Assume D is invertible and abs znormD^-1<1. Then Σ(z) =- ∑_n=0^∞ z^nBD^-n-1B^* with norm convergence. In particular, F_P(z) = z-A+BD^-1B^*+O(abs znorm B^2normD^-1^2).
Exact mathematical passage
F_P(z) = z-A+BD^{-1}B^*+O\bigl(\abs z\norm B^2\norm{D^{-1}}^2\bigr).
b_a^{\phantom a c}Use in source · 1 passage
Search aliases: b_a^phantom a c phantom
Use in source · Read passage · 02_strengthened.tex, line 2241
[Invariance of the physical space] Let H=H^*. Suppose [H,C_a] = ∑_cb_a^phantom a cC_c for every a. Then H preserves H_phys, and so does e^- i tH for every real t.
Exact mathematical passage
[H,C_a] = \sum_cb_a^{\phantom a c}C_c
B_iUse in source · 1 passage
Search aliases: B_i
Use in source · Read passage · 03_supplement_02.tex, line 71
Expanding the finite product of sums gives T(A) T(B)= T(i↦ A_iB_i). Consequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so Q_a^*=Q_a. Summing over all address functions and using the same expansion yields ∑_a Q_a= T(i↦∑_jP^(i)_j)=I. A separate algebraic regrouping theorem applies to any finite orthogonal family and a…
Exact mathematical passage
\mathsf T(A)\mathsf T(B)=\mathsf T(i\mapsto A_iB_i).
B_rUse in source · 5 passages
Search aliases: B_r
Use in source · Read passage · 03_supplement_05.tex, line 49
Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.
Exact mathematical passage
B_r=\sum_a U_{ar}F_a.
B_r^*Use in source · 2 passages
Search aliases: B_r^*
Use in source · Read passage · 03_supplement_05.tex, line 53
Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.
Exact mathematical passage
B_rP=B_r,\qquad B_r^*B_s=\delta_{rs}\lambda_rP, \qquad \sum_rB_r\rho B_r^*=\sum_aF_a\rho F_a^*.
B_sUse in source · 1 passage
Search aliases: B_s
Use in source · Read passage · 03_supplement_05.tex, line 53
Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.
Exact mathematical passage
B_rP=B_r,\qquad B_r^*B_s=\delta_{rs}\lambda_rP, \qquad \sum_rB_r\rho B_r^*=\sum_aF_a\rho F_a^*.
\bar bUse in source · 1 passage
Search aliases: bar b bar
Use in source · monograph_atlas.tex, line 156
ity. Starting from ket0, there are two resolved paths to output ket1. Their signed products are bra1H_rm Hadket0bra0H_rm Hadket0=frac12, bra1H_rm Hadket1bra1H_rm Hadket0=-frac12. Their sum is zero. Squaring each separately gives a different answer. vfiginterference headWhere the missing term goes For path amplitudes a,b, the outcome probability is |a+b|^2=|a|^2+|b|^2+2Re(abar b). Here the last term is -1/2, canceling the first two terms. The path picture retains this information only if its edge…
Exact mathematical passage
|a+b|^2=|a|^2+|b|^2+2\Re(a\bar b)
\bbCUse in source · 11 passages
Search aliases: bbC C mathbb
Use in source · Read passage · 01_finite_core.tex, line 79
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cA \cong \bigoplus_{i=1}^r \left(M_{d_i}(\bbC)\otimes I_{m_i}\right)
\bbC^dUse in source · 1 passage
Search aliases: bbC^d C^d bbC mathbb
Use in source · Read passage · 01_finite_core.tex, line 95
[Orbit dimension of one quantum seed] For integers d≥1 and m≥0, consider the action of M_d( C)⊗ I_m on C^d⊗ C^m. Let C be the coefficient matrix of a vector ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} =d rank C. The seed is cyclic if and only if rank C=m, which is possible only when m≤ d.
Exact mathematical passage
\bbC^d\otimes\bbC^m
\bbC^{d_i}Use in source · 3 passages
Search aliases: bbC^d_i C^d_i bbC mathbb
Use in source · Read passage · 01_finite_core.tex, line 86
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cH \cong \bigoplus_{i=1}^r \left(\bbC^{d_i}\otimes\bbC^{m_i}\right).
\bbC^mUse in source · 1 passage
Search aliases: bbC^m C^m bbC mathbb
Use in source · Read passage · 01_finite_core.tex, line 95
[Orbit dimension of one quantum seed] For integers d≥1 and m≥0, consider the action of M_d( C)⊗ I_m on C^d⊗ C^m. Let C be the coefficient matrix of a vector ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} =d rank C. The seed is cyclic if and only if rank C=m, which is possible only when m≤ d.
Exact mathematical passage
\bbC^d\otimes\bbC^m
\bbC^{m_i}Use in source · 6 passages
Search aliases: bbC^m_i C^m_i bbC mathbb
Use in source · Read passage · 01_finite_core.tex, line 86
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cH \cong \bigoplus_{i=1}^r \left(\bbC^{d_i}\otimes\bbC^{m_i}\right).
\bbRUse in source · 2 passages
Search aliases: bbR R mathbb
Use in source · Read passage · 01_finite_core.tex, line 573
[Projected connection] Let M be a smooth finite-dimensional real manifold without boundary, let V be a finite-dimensional real normed vector space, and let P:M→End_ R(V) be smooth with P(x)^2=P(x). In the quantum application, V is the underlying real space of a finite complex Hilbert space and P is an orthogonal complex-linear projector.
Exact mathematical passage
P:M\to\operatorname{End}_{\bbR}(V)
\bbZUse in source · 1 passage
Search aliases: bbZ Z mathbb
Use in source · Read passage · 01_finite_core.tex, line 246
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
r\in\bbZ/N\bbZ
\bmodUse in source · 2 passages
Search aliases: bmod
Use in source · monograph_atlas.tex, line 92
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing…
Exact mathematical passage
S\ket k=\ket{k+1\bmod4}
\braUse in source · 21 passages
Search aliases: bra
Use in source · monograph_atlas.tex, line 108
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigpar…
Exact mathematical passage
P_a=\ket a\bra a
\mathbb CUse in source · 2 passages
Search aliases: C mathbb
Use in source · Read passage · monograph_examples.tex, line 3
Three exact finite models. addcontentslinetocsectionThree exact finite models A seed remembers multiplicity support. For M_2( C)⊗ I_3, a coefficient matrix with rank one generates an orbit of dimension two. A matrix of rank two generates an orbit of dimension four. Neither single seed spans the full space of dimension six, because a 2×3 matrix cannot have rank three. Two seeds can nevertheless cover all three multiplicity coordinates together. The example illustrates the finite seed rank formula…
Exact mathematical passage
M_2(\mathbb C)\otimes I_3
c^*Use in source · 1 passage
Search aliases: c^*
Use in source · Read passage · 03_supplement_05.tex, line 175
For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.
Exact mathematical passage
\mathcal R\bigl(E(c)\rho E(c)^*\bigr) =\left(\sum_{a,b}c_a\overline{c_b}C_{ba}\right)\rho =(c^*Cc)\rho.
C^2Use in source · 2 passages
Search aliases: C^2
Use in source · Read passage · 03_supplement_04.tex, line 49
Actual projected differentiation and curvature. Let E,F be real normed spaces. Let P:E→ L(F,F) be differentiable with P(x)^2=P(x) for every x. Let s:E→ F be C^2 with P(x)s(x)=s(x), and let X,Y:E→ E be differentiable vector fields. The checked connection is the actual expression ∇_Xs(x)=P(x) Ds(x)[X(x)]. It is proved to preserve the range condition. It is linear in the vector field direction and obeys the scalar Leibniz rule ∇_X(cs)=Dc[X] s+c∇_Xs for differentiable real scalar functions c. No dif…
Exact mathematical passage
C^2
\mathbb C^2Use in source · 1 passage
Search aliases: C^2 mathbb
Use in source · Read passage · monograph_examples.tex, line 17
model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The …
Exact mathematical passage
\mathbb C^2
\C^nUse in source · 1 passage
Search aliases: C^n C mathbb
Use in source · Read passage · 03_supplement_05.tex, line 2
The precise finite channel statement. Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^*R_r=I. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire …
Exact mathematical passage
\mathcal H=\C^n
C_1Use in source · 14 passages
Search aliases: C_1
Use in source · monograph_atlas.tex, line 75
vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0 0 · 1 · 0, C_2=mat0 · 0 · 1 0 · 0 · 0. These two matrices provide a complete set of row directions. Their individual orbit dimensions are four and two, and …
Exact mathematical passage
C_1=\mat{1&0&0\\0&1&0},\qquad C_2=\mat{0&0&1\\0&0&0}.
C_2Use in source · 11 passages
Search aliases: C_2
Use in source · monograph_atlas.tex, line 75
vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0 0 · 1 · 0, C_2=mat0 · 0 · 1 0 · 0 · 0. These two matrices provide a complete set of row directions. Their individual orbit dimensions are four and two, and …
Exact mathematical passage
C_1=\mat{1&0&0\\0&1&0},\qquad C_2=\mat{0&0&1\\0&0&0}.
c_aUse in source · 2 passages
Search aliases: c_a
Use in source · Read passage · 03_supplement_05.tex, line 174
For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.
Exact mathematical passage
E(c)=\sum_a c_aE_a
C_{ab}Use in source · 3 passages
Search aliases: C_ab
Use in source · Read passage · 03_supplement_05.tex, line 14
Necessary and sufficient conditions. Under the stated normalization, the following equivalence is proved: ∃ R of the above finite Kraus form, [-2pt] R N_E(ρ)=ρ for every ρsucceq0, Trρ=1, Pρ P=ρ Longleftrightarrow ∃ Csucceq0, PE_a^*E_bP=C_abP for all a,b. The statement applies throughout the code space. The formalization separately proves equivalence with correction of every complex matrix supported on the code. It also proves preservation with arbitrary finite reference systems, as detailed in t…
Exact mathematical passage
\boxed{\begin{gathered} \exists\,\mathcal R\text{ of the above finite Kraus form,}\\[-2pt] \mathcal R\mathcal N_E(\rho)=\rho \quad\text{for every }\rho\succeq0,\ \Tr\rho=1,\ P\rho P=\rho \end{gathered}} \quad\Longleftrightarrow\quad \boxed{\begin{gathered} \exists C\succeq0,\\ PE_a^*E_bP=C_{ab}P\quad\text{for all }a,b. \end{gathered}}
c_{ab}Use in source · 8 passages
Search aliases: c_ab
Use in source · monograph_atlas.tex, line 598
vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATATraditional edition: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0 12 · 16 · 0 0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · …
Exact mathematical passage
c=\frac1{25}\mat{9&12&0\\12&16&0\\0&0&0}, \qquad PE_a^*E_bP=c_{ab}P.
C_{ba}Use in source · 2 passages
Search aliases: C_ba
Use in source · Read passage · 03_supplement_05.tex, line 169
Cross terms for the original errors. For arbitrary positive Gram data, the same recovery satisfies R(E_aρ E_b^*)=C_baρ (Pρ P=ρ). The index order ba is intentional: the defining Gram relation is PE_a^*E_bP=C_abP. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to identify the original Gram entries.
Exact mathematical passage
\boxed{\mathcal R(E_a\rho E_b^*)=C_{ba}\rho\qquad(P\rho P=\rho).}
C_iUse in source · 1 passage
Search aliases: C_i
Use in source · Read passage · 01_finite_core.tex, line 92
lus_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
C_i
C_{ij}Use in source · 2 passages
Search aliases: C_ij
Use in source · Read passage · 03_supplement_03.tex, line 45
Schmidt rank and the reduced state. Use coordinates ψ_ij in the standard product basis and let C_ij=ψ_ij. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action: coeff((A⊗ I_B)ψ)=AC. It also proves that the transported matrix orbit equals the span of the state vectors obtained by this Kronecker action. The dimension formula therefore applies to the physical local action on the state sp…
Exact mathematical passage
C_{ij}=\psi_{ij}
c_{ja}Use in source · 1 passage
Search aliases: c_ja
Use in source · Read passage · 03_supplement_05.tex, line 183
Every normalized channel in the same span. Suppose F_j=∑_a c_jaE_a and ∑_jF_j^*F_j=I. The decoder is not reconstructed for the F_j. Summing the coherent identities gives a scalar multiple qρ. Trace preservation of both the F channel and the recovery forces q=1 on a nonzero code. The proof uses positivity of P to justify cancellation of its trace and handles P=0 separately. Therefore R(∑_jF_jρ F_j^*)=ρ. This is pathData.recovery_span_channel, a quantified statement over arbitrary finite F familie…
Exact mathematical passage
F_j=\sum_a c_{ja}E_a
C_{jk}Use in source · 1 passage
Search aliases: C_jk
Use in source · Read passage · 02_strengthened.tex, line 1903
[Reflection positive correlators] Fix a vector Ω and operators A_1,…,A_m. The matrix C_jk = ipA_jΩT^n_j+n_kA_kΩ is positive semidefinite.
Exact mathematical passage
C_{jk} = \ip{A_j\Omega}{T^{n_j+n_k}A_k\Omega}
C_kUse in source · 1 passage
Search aliases: C_k
Use in source · Read passage · 03_supplement_03.tex, line 17
Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and proves the equivalence the seeds generate Longleftrightarrow rank D=m. Thus m≤ kd is necessary. It is sufficient as well: embed the m coordinate columns i…
Exact mathematical passage
C_1,\ldots,C_k
c_kUse in source · 2 passages
Search aliases: c_k
Use in source · Read passage · 02_strengthened.tex, line 1880
[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.
Exact mathematical passage
\sum_{j,k}\overline{c_j}c_k \ip{v_j}{T^{n_j+n_k}v_k} \geq0
C_mUse in source · 2 passages
Search aliases: C_m
Use in source · Read passage · 02_strengthened.tex, line 2174
Let C_1,…,C_m be self adjoint constraint operators on a finite dimensional Hilbert space. The distinction between constraint action and constraint closure is standard in constrained Hamiltonian theory [source reference]. Write c=span_ C{C_1,…,C_m} ⊆End( H).
Exact mathematical passage
C_1,\ldots,C_m
C_nUse in source · 5 passages
Search aliases: C_n
Use in source · monograph_atlas.tex, line 788
vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPETraditional edition: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how …
Exact mathematical passage
C_n(j,k)=1+q_n^{j+k}
C_{n,a}Use in source · 2 passages
Search aliases: C_n,a
Use in source · Read passage · 02_strengthened.tex, line 2491
[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.
Exact mathematical passage
C_{n,a}\to C_a
C_{n,b}Use in source · 1 passage
Search aliases: C_n,b
Use in source · Read passage · 02_strengthened.tex, line 2492
[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.
Exact mathematical passage
\norm{ [C_{n,a},C_{n,b}] -\ii\sum_cf_{ab}^{\phantom{ab}c}C_{n,c} } \longrightarrow0,
C_{n,c}Use in source · 1 passage
Search aliases: C_n,c
Use in source · Read passage · 02_strengthened.tex, line 2492
[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.
Exact mathematical passage
\norm{ [C_{n,a},C_{n,b}] -\ii\sum_cf_{ab}^{\phantom{ab}c}C_{n,c} } \longrightarrow0,
C_rUse in source · 3 passages
Search aliases: C_r
Use in source · Read passage · 03_supplement_02.tex, line 54
Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.
Exact mathematical passage
Q_a=\mathsf T\bigl(i\mapsto P^{(i)}_{a_i}\bigr), \qquad C_r=\sum_{\sum_i a_i=r}Q_a.
C_r^*Use in source · 1 passage
Search aliases: C_r^*
Use in source · Read passage · 03_supplement_02.tex, line 62
The fine and coarse families are complete orthogonal self-adjoint projector systems. In particular, C_rC_s=δ_rsC_r, ∑_r C_r=I, C_r^*=C_r. Each coarse projector is exactly the displayed sum of its fine descendants.
Exact mathematical passage
C_rC_s=\delta_{rs}C_r,\qquad \sum_r C_r=I,\qquad C_r^*=C_r.
C_sUse in source · 2 passages
Search aliases: C_s
Use in source · Read passage · 03_supplement_02.tex, line 62
The fine and coarse families are complete orthogonal self-adjoint projector systems. In particular, C_rC_s=δ_rsC_r, ∑_r C_r=I, C_r^*=C_r. Each coarse projector is exactly the displayed sum of its fine descendants.
Exact mathematical passage
C_rC_s=\delta_{rs}C_r,\qquad \sum_r C_r=I,\qquad C_r^*=C_r.
C_TUse in source · 1 passage
Search aliases: C_T
Use in source · Read passage · 02_strengthened.tex, line 812
em[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
\norm{W_\pi-W(T)} \leq C_T\abs{\pi}
C_xUse in source · 1 passage
Search aliases: C_x
Use in source · monograph_atlas.tex, line 747
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical ke…
Exact mathematical passage
[C_x,C_y]=2\ii C_z,\qquad[C_y,C_z]=2\ii C_x,\qquad[C_z,C_x]=2\ii C_y.
C_yUse in source · 1 passage
Search aliases: C_y
Use in source · monograph_atlas.tex, line 747
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical ke…
Exact mathematical passage
[C_x,C_y]=2\ii C_z,\qquad[C_y,C_z]=2\ii C_x,\qquad[C_z,C_x]=2\ii C_y.
C_zUse in source · 1 passage
Search aliases: C_z
Use in source · monograph_atlas.tex, line 747
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical ke…
Exact mathematical passage
[C_x,C_y]=2\ii C_z,\qquad[C_y,C_z]=2\ii C_x,\qquad[C_z,C_x]=2\ii C_y.
C_\psiUse in source · 10 passages
Search aliases: C_ψ psi
Use in source · Read passage · 02_strengthened.tex, line 120
[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.
Exact mathematical passage
C_\psi
C_\ellUse in source · 7 passages
Search aliases: C_ℓ ell
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
C_{\ell+1,a}Use in source · 10 passages
Search aliases: C_ℓ+1,a ell
Use in source · Read passage · 02_strengthened.tex, line 2261
Let ( H_ℓ,C_ℓ,a) and ( H_ℓ+1,C_ℓ+1,a) be two levels, and let R_ℓ: H_ℓ⟶ H_ℓ+1 be an isometry.
Exact mathematical passage
(\Hcal_{\ell+1},C_{\ell+1,a})
C_{\ell+1,b}Use in source · 3 passages
Search aliases: C_ℓ+1,b ell
Use in source · Read passage · 02_strengthened.tex, line 2273
[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.
Exact mathematical passage
[C_{\ell+1,a},C_{\ell+1,b}]R_\ell = R_\ell[C_{\ell,a},C_{\ell,b}].
C_{\ell,a}Use in source · 10 passages
Search aliases: C_ℓ,a ell
Use in source · Read passage · 02_strengthened.tex, line 2261
Let ( H_ℓ,C_ℓ,a) and ( H_ℓ+1,C_ℓ+1,a) be two levels, and let R_ℓ: H_ℓ⟶ H_ℓ+1 be an isometry.
Exact mathematical passage
(\Hcal_\ell,C_{\ell,a})
C_{\ell,b}Use in source · 3 passages
Search aliases: C_ℓ,b ell
Use in source · Read passage · 02_strengthened.tex, line 2273
[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.
Exact mathematical passage
[C_{\ell+1,a},C_{\ell+1,b}]R_\ell = R_\ell[C_{\ell,a},C_{\ell,b}].
\cAUse in source · 5 passages
Search aliases: cA A mathcal
Use in source · Read passage · 01_finite_core.tex, line 79
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cA \cong \bigoplus_{i=1}^r \left(M_{d_i}(\bbC)\otimes I_{m_i}\right)
\cA_\ellUse in source · 1 passage
Search aliases: cA_ℓ A_ℓ cA ell mathcal ell
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\cCUse in source · 3 passages
Search aliases: cC C mathcal
Use in source · Read passage · 01_finite_core.tex, line 792
Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environment can distinguish logical states and exact correction is impossible.
Exact mathematical passage
\cC\subset\cH
\CC^3Use in source · 9 passages
Search aliases: CC^3 C^3 CC mathbb
Use in source · monograph_atlas.tex, line 74
vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0
Exact mathematical passage
\CC^2\otimes\CC^3
\CC^4Use in source · 1 passage
Search aliases: CC^4 C^4 CC mathbb
Use in source · monograph_atlas.tex, line 753
l is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physica…
Exact mathematical passage
R:\CC^3\to\CC^4
\CC^6Use in source · 1 passage
Search aliases: CC^6 C^6 CC mathbb
Use in source · monograph_atlas.tex, line 86
o groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines the required count: srank_AA(HH)=max_ilceil m_i/d_irceil. This is an invariant of the declared representation of the algebra, and depends on more than the dimension of the Hilbert space alone. If the available algebra were enlarged to the full matrix algebra on C^6, every nonzero seed would instead be cyclic.
Exact mathematical passage
\CC^6
\CC^8Use in source · 1 passage
Search aliases: CC^8 C^8 CC mathbb
Use in source · monograph_atlas.tex, line 52
2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'' The eight vectors are orthonormal, so their linear span is all of C^8. This proves cyclicity for the algebra generated by the flips. Preparing every normalized vector requires more than products of these fl…
Exact mathematical passage
\CC^8
\CC^dUse in source · 1 passage
Search aliases: CC^d C^d CC mathbb
Use in source · Read passage · 02_strengthened.tex, line 120
[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.
Exact mathematical passage
\CC^d\otimes\CC^m
\CC^{d_r}Use in source · 1 passage
Search aliases: CC^d_r C^d_r CC mathbb
Use in source · Read passage · 02_strengthened.tex, line 111
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Hcal \cong \bigoplus_{r=1}^{s} \bigl(\CC^{d_r}\otimes\CC^{m_r}\bigr).
\CC^mUse in source · 2 passages
Search aliases: CC^m C^m CC mathbb
Use in source · Read passage · 02_strengthened.tex, line 120
[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.
Exact mathematical passage
\CC^d\otimes\CC^m
\CC^{m_r}Use in source · 2 passages
Search aliases: CC^m_r C^m_r CC mathbb
Use in source · Read passage · 02_strengthened.tex, line 111
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Hcal \cong \bigoplus_{r=1}^{s} \bigl(\CC^{d_r}\otimes\CC^{m_r}\bigr).
\CC^nUse in source · 2 passages
Search aliases: CC^n C^n CC mathbb
Use in source · monograph_atlas.tex, line 765
vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and…
Exact mathematical passage
\HH_n=\CC^n
\CC^{n+1}Use in source · 1 passage
Search aliases: CC^n+1 C^n+1 CC mathbb
Use in source · tower_tail.tex, line 5
H_n=diag(0,1,1/2,…); node[draw,fill=black!7,align=center] (b) at (9.5,2.2) C^n+1
Exact mathematical passage
\CC^{n+1}
\CC^rUse in source · 2 passages
Search aliases: CC^r C^r CC mathbb
Use in source · Read passage · 02_strengthened.tex, line 851
Let P(x) be a smooth rank r orthogonal projector over a parameter manifold. Choose a local isometry V(x): C^r⟶ H, V(x)V(x)^*=P(x).
Exact mathematical passage
V(x):\CC^r\longrightarrow\Hcal, \qquad V(x)V(x)^*=P(x).
\cDUse in source · 2 passages
Search aliases: cD D mathcal
Use in source · Read passage · 01_finite_core.tex, line 691
[Block-diagonal measurement invariance] Let {P_α} be a complete packet family, and suppose every measurement effect is packet diagonal: E_k=∑_α P_α E_kP_α. Define the packet-dephased state D_P(ρ)=∑_α P_αρ P_α. Then Tr(E_kρ) = Tr(E_k D_P(ρ)) for every outcome k.
Exact mathematical passage
\cD_P(\rho)=\sum_\alpha P_\alpha\rho P_\alpha.
\cD_PUse in source · 2 passages
Search aliases: cD_P D_P cD mathcal
Use in source · Read passage · 01_finite_core.tex, line 691
[Block-diagonal measurement invariance] Let {P_α} be a complete packet family, and suppose every measurement effect is packet diagonal: E_k=∑_α P_α E_kP_α. Define the packet-dephased state D_P(ρ)=∑_α P_αρ P_α. Then Tr(E_kρ) = Tr(E_k D_P(ρ)) for every outcome k.
Exact mathematical passage
\cD_P(\rho)=\sum_\alpha P_\alpha\rho P_\alpha.
\cFUse in source · 2 passages
Search aliases: cF F mathcal
Use in source · Read passage · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\cF_{\theta\phi}Use in source · 1 passage
Search aliases: cF_θϕ F_θϕ cF theta phi mathcal theta phi
Use in source · Read passage · 01_finite_core.tex, line 652
I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.
Exact mathematical passage
\boxed{\cF_{\theta\phi} =\frac12\mathbf n\cdot(\partial_\theta\mathbf n\mathbin{\times}\partial_\phi\mathbf n) =\frac12\sin\theta.}
\cF_{\mu\nu}Use in source · 1 passage
Search aliases: cF_μν F_μν cF mu nu mathcal mu nu
Use in source · Read passage · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\cH_AUse in source · 2 passages
Search aliases: cH_A H_A cH mathcal
Use in source · Read passage · 01_finite_core.tex, line 137
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\psi\in\cH_A\otimes\cH_B
\cH_BUse in source · 2 passages
Search aliases: cH_B H_B cH mathcal
Use in source · Read passage · 01_finite_core.tex, line 137
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\psi\in\cH_A\otimes\cH_B
\cH_\ellUse in source · 3 passages
Search aliases: cH_ℓ H_ℓ cH ell mathcal ell
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\cH_{\ell+1}Use in source · 2 passages
Search aliases: cH_ℓ+1 H_ℓ+1 cH ell mathcal ell
Use in source · Read passage · 01_finite_core.tex, line 1019
hfrak Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
E_\ell:\cH_\ell\to\cH_{\ell+1}
\mathrm{CNOT}Use in source · 1 passage
Search aliases: CNOT mathrm
Use in source · monograph_atlas.tex, line 136
le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original 4×4 circuit matrix: ∑_a,b,cP_b CNOT P_a (H_rm Had⊗ I)P_c =CNOT(H_rm Had⊗ I). This expansion is an identity even though the intermediate parity …
Exact mathematical passage
\sum_{a,b,c}P_b\,\mathrm{CNOT}\,P_a (H_{\rm Had}\otimes I)P_c =\mathrm{CNOT}(H_{\rm Had}\otimes I).
\mathrm{coarse}Use in source · 2 passages
Search aliases: coarse mathrm
Use in source · Read passage · 01_finite_core.tex, line 247
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
\boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. }
\operatorname{coeff}Use in source · 1 passage
Search aliases: coeff operatorname
Use in source · Read passage · 03_supplement_03.tex, line 46
Schmidt rank and the reduced state. Use coordinates ψ_ij in the standard product basis and let C_ij=ψ_ij. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action: coeff((A⊗ I_B)ψ)=AC. It also proves that the transported matrix orbit equals the span of the state vectors obtained by this Kronecker action. The dimension formula therefore applies to the physical local action on the state sp…
Exact mathematical passage
\operatorname{coeff}\bigl((A\otimes I_B)\psi\bigr)=AC.
\commUse in source · 2 passages
Search aliases: comm
Use in source · Read passage · 03_supplement_02.tex, line 146
Define C=∑_aD_aP_a(t). Then C=frac12∑_acommD_aP_a(t), C^*=-C, and, for every b, commCP_b(t)=D_b.
Exact mathematical passage
\boxed{\quad C=\frac12\sum_a\comm{D_a}{P_a(t)},\qquad C^*=-C,\quad}
\cosUse in source · 10 passages
Search aliases: cos
Use in source · monograph_atlas.tex, line 188
vpage11Quantify the amplitude that escapesCOMMUTATORS AND PROBABILITY BOUNDSTraditional edition: Theorem II.5.3, Corollary II.5.4. For a two-level system, choose H=ε X, P=ket0bra0, and Q=I-P. Direct exponentiation gives e^- i tH=cos(ε t)I- isin(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probabi…
Exact mathematical passage
e^{-\ii tH}=\cos(\varepsilon t)I-\ii\sin(\varepsilon t)X, \quad \norm{Qe^{-\ii tH}P}=|\sin(\varepsilon t)|, \quad\norm{[H,P]}=|\varepsilon|.
\cos^2Use in source · 1 passage
Search aliases: cos^2 cos
Use in source · monograph_atlas.tex, line 261
E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). It equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates. vfigrange_atlas headAn exact example of the domain restriction For the rotating line and anchor P(0), T_0(t)=cos t U(t), det T_0(t)=cos^2t. The chart fails at t=π/2, but P(π/2)=diag(0,1) is a perfectly regular rank-one proje…
Exact mathematical passage
T_0(t)=\cos t\,U(t),\qquad \det T_0(t)=\cos^2t.
D^*Use in source · 1 passage
Search aliases: D^*
Use in source · Read passage · 02_strengthened.tex, line 1786
[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D) ≥δ>0. Then normΣ(z)-widetildeΣ(z) ≤ frac2gδnormB-widetilde B + fracg^2δ^2normD-widetilde D.
Exact mathematical passage
D=D^*
D^{-1}Use in source · 4 passages
Search aliases: D^-1
Use in source · monograph_atlas.tex, line 669
Sigma|=|1-frac8156|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings. headA series near zero energy and its exact remainder At z=1/2, |z| |D^-1|=1/4<1, so Σ(z)=-∑_n=0^∞fracz^n2^n+1. After the first four terms, the exact remainder has magnitude 1/384. The geometric series controls the infinite tail…
Exact mathematical passage
|z|\,|D^{-1}|=1/4<1
D^{-n-1}Use in source · 2 passages
Search aliases: D^-n-1
Use in source · Read passage · 02_strengthened.tex, line 1633
[Low energy expansion] Assume D is invertible and abs znormD^-1<1. Then Σ(z) =- ∑_n=0^∞ z^nBD^-n-1B^* with norm convergence. In particular, F_P(z) = z-A+BD^-1B^*+O(abs znorm B^2normD^-1^2).
Exact mathematical passage
\Sigma(z) =- \sum_{n=0}^{\infty} z^nBD^{-n-1}B^*
D^2Use in source · 1 passage
Search aliases: D^2
Use in source · Read passage · 02_strengthened.tex, line 1953
A finite spectral datum consists of a finite dimensional Hilbert space, a nonnegative self adjoint operator L, and any declared packet family. One may take L=D^2 for a finite Dirac type operator or a graph Laplacian. Related geometric viewpoints appear in [source reference].
Exact mathematical passage
L=D^2
d^4Use in source · 1 passage
Search aliases: d^4
Use in source · Read passage · 01_finite_core.tex, line 1240
[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.
Exact mathematical passage
\frac1{16\pi G} \int\sqrt{-g}\,(R-2\Lambda)\,d^4x
D_aUse in source · 7 passages
Search aliases: D_a
Use in source · Read passage · 03_supplement_02.tex, line 138
Let P_a(s) be a finite family in a normed real algebra with a continuous compatible adjoint. Assume actual derivatives D_a=dot P_a(t) exist and, for all parameter values under consideration, P_aP_b=δ_abP_a, ∑_aP_a=I, P_a^*=P_a. In the supplied Lean theorem, the curves are defined on R and the projector laws hold at every real parameter. The algebraic formulation applies to matrices and bounded operators whenever they satisfy the stated typeclass hypotheses.
Exact mathematical passage
D_a=\dot P_a(t)
d_AUse in source · 8 passages
Search aliases: d_A
Use in source · Read passage · 01_finite_core.tex, line 137
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\dim\cH_A=d_A\ge1
D_a^*Use in source · 1 passage
Search aliases: D_a^*
Use in source · Read passage · 03_supplement_02.tex, line 161
The derivative product rule and uniqueness of derivatives give D_aP_b+P_aD_b=δ_abD_a. Differentiating completeness and adjoint invariance gives ∑_aD_a=0, D_a^*=D_a. These are proved derivative consequences, not additional axioms supplied in place of differentiation.
Exact mathematical passage
\sum_aD_a=0,\qquad D_a^*=D_a.
D_bUse in source · 5 passages
Search aliases: D_b
Use in source · Read passage · 03_supplement_02.tex, line 150
Define C=∑_aD_aP_a(t). Then C=frac12∑_acommD_aP_a(t), C^*=-C, and, for every b, commCP_b(t)=D_b.
Exact mathematical passage
\boxed{\comm{C}{P_b(t)}=D_b.}
D_iUse in source · 1 passage
Search aliases: D_i
Use in source · Read passage · 03_supplement_02.tex, line 47
Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite number of factors with the stated finite coordinate sets.
Exact mathematical passage
D_i
d_{ja}Use in source · 1 passage
Search aliases: d_ja
Use in source · Read passage · 01_finite_core.tex, line 829
ry vector orthogonal to ψ, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recovery completeness now gives PE_a^*E_bP · =∑_j (K_jE_aP)^*(K_jE_bP)
Exact mathematical passage
K_jE_aP=d_{ja}P.
d_{jb}Use in source · 1 passage
Search aliases: d_jb
Use in source · Read passage · 01_finite_core.tex, line 833
he left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recovery completeness now gives PE_a^*E_bP · =∑_j (K_jE_aP)^*(K_jE_bP) · =(∑_joverlined_jad_jb)P. The coefficients form a Gram matrix, hence are positive se…
Exact mathematical passage
\begin{aligned} PE_a^*E_bP &=\sum_j (K_jE_aP)^*(K_jE_bP)\\ &=\left(\sum_j\overline{d_{ja}}d_{jb}\right)P. \end{aligned}
d_{L,a}Use in source · 2 passages
Search aliases: d_L,a
Use in source · Read passage · 02_strengthened.tex, line 1999
[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.
Exact mathematical passage
d_L(t) = \sum_{a:Z_{L,a}(t)>0} \frac{Z_{L,a}(t)}{Z_L(t)} d_{L,a}(t),
d_{L_N}Use in source · 1 passage
Search aliases: d_L_N
Use in source · Read passage · 02_strengthened.tex, line 2153
[Rigorous intermediate dimension window] For every ε>0, there is a diffusion time t_ε>0 and an integer N_ε such that absd_L_N(t_ε)-d<ε for every N≥ N_ε.
Exact mathematical passage
\abs{d_{L_N}(t_\varepsilon)-d}<\varepsilon
d_NUse in source · 4 passages
Search aliases: d_N
Use in source · monograph_atlas.tex, line 725
vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSTraditional edition: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).
Exact mathematical passage
d_N(t)=2t\sum_k\lambda_N(k)e^{-t\lambda_N(k)}/Z_N(t)
d_rUse in source · 9 passages
Search aliases: d_r
Use in source · Read passage · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
D_{\rm computational}Use in source · 2 passages
Search aliases: D_rm computational rm
Use in source · monograph_atlas.tex, line 299
g Every effect in the computational basis E_a=ket abra a has the same expectation on both matrices, namely 1/2 at 00 and 11 and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture. headA measurement that resolves the difference Use the valid two-outcome measurement with effects E=ρ and I-E. Then Tr(Eρ)=1, Tr (E D_rm computational(ρ))=frac12. The effect itself is not diagonal in the fine computational packets. It comp…
Exact mathematical passage
\Tr(E\rho)=1,\qquad \Tr\!\left(E D_{\rm computational}(\rho)\right)=\frac12.
D_{\rm parity}Use in source · 1 passage
Search aliases: D_rm parity rm
Use in source · dephasing.tex, line 3
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (2.25,4.65) 2ρ, also 2D_rm parity(ρ); path[draw=black!30,fill=black!8] (0.75,3.2) rectangle (1.5,3.95); node[font=small,align=center,font=small] at (1.125,3.575) 1; path[draw=black!30,fill=white] (1.5,3.2) rectangle (2.25,3.95); node[font=small,align=center,font=small] at (1.875,3.575) 0; path[draw=black!30,fill=white] (2.25,3.2) rectangle (3,3.…
Exact mathematical passage
2D_{\rm parity}(\rho)
d_{\rm torus}Use in source · 3 passages
Search aliases: d_rm torus rm
Use in source · Read passage · 02_strengthened.tex, line 2120
[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.
Exact mathematical passage
d_{\rm torus}(t) =-2\frac{\dd\log Z_d(t)}{\dd\log t}.
d_sUse in source · 6 passages
Search aliases: d_s
Use in source · monograph_atlas.tex, line 442
vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESTraditional edition: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect…
Exact mathematical passage
K(\tau)=\frac{1+2e^{-2\tau}+e^{-4\tau}}4 =\frac{(1+e^{-2\tau})^2}4, \qquad d_s(\tau)=-2\tau\frac{K'(\tau)}{K(\tau)} =\frac{8\tau}{e^{2\tau}+1}.
d_{s,2}Use in source · 1 passage
Search aliases: d_s,2
Use in source · monograph_atlas.tex, line 454
For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.
Exact mathematical passage
d_{s,2}=2d_s
d_s^{(1)}Use in source · 1 passage
Search aliases: d_s^(1)
Use in source · Read passage · 01_finite_core.tex, line 1188
[Spectral dimension] The median intermediate plateaus were d_s^(1)=1.00062, d_s^(2)=2.03781. These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.
Exact mathematical passage
d_s^{(1)}=1.00062, \qquad d_s^{(2)}=2.03781.
d_s^{(2)}Use in source · 1 passage
Search aliases: d_s^(2)
Use in source · Read passage · 01_finite_core.tex, line 1188
[Spectral dimension] The median intermediate plateaus were d_s^(1)=1.00062, d_s^(2)=2.03781. These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.
Exact mathematical passage
d_s^{(1)}=1.00062, \qquad d_s^{(2)}=2.03781.
D_zUse in source · 5 passages
Search aliases: D_z
Use in source · Read passage · 03_supplement_01.tex, line 106
-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).
Exact mathematical passage
D_z=zI_Q-D,\qquad S_z=zI_P-A-BD_z^{-1}C.
D_z^{-1}Use in source · 2 passages
Search aliases: D_z^-1
Use in source · Read passage · 03_supplement_01.tex, line 106
-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).
Exact mathematical passage
D_z=zI_Q-D,\qquad S_z=zI_P-A-BD_z^{-1}C.
D_\ellUse in source · 2 passages
Search aliases: D_ℓ ell
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
D_{\ell+1}Use in source · 1 passage
Search aliases: D_ℓ+1 ell
Use in source · Read passage · 01_finite_core.tex, line 1038
A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Exact mathematical passage
\Delta_\ell(D) =C_\ell D_{\ell+1}E_\ell-D_\ell.
\ddot xUse in source · 4 passages
Search aliases: ddot x ddot
Use in source · monograph_atlas.tex, line 409
returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict x(t)=1. vfigmemory headEliminate exactly, not approximately Integrating the second equation with y(0)=0 gives y(t)=- i∫_0^tx(s) ds. Substitution produces dot x(t)=-∫_0^tx(s) ds. The memory kernel is the constant function one. Differentiating once more recovers ddot x=-x with x(0)=1 and dot x(0)=0, explaining the cosine solution. A nonzero hidden initial state would add a forcing term tha…
Exact mathematical passage
\ddot x=-x
\dephUse in source · 4 passages
Search aliases: deph D_P mathcal
Use in source · Read passage · 03_supplement_01.tex, line 75
Measurements: source Theorem 7.1. For finite complex matrices, define packet dephasing by D_P(X)=∑_aP_aXP_a. The code proves idempotence and trace duality: D_P( D_P(X))= D_P(X), tr( D_P(E)ρ)=tr(E D_P(ρ)). Trace duality follows term by term from cyclicity: tr(P_aEP_aρ)=tr(EP_aρ P_a). Therefore an effect that is diagonal in the packets, D_P(E)=E, satisfies tr(Eρ)=tr(E D_P(ρ)). The proof holds for arbitrary finite complex matrices E and ρ. Density matrices and positive measurement effects are valid…
Exact mathematical passage
\deph(X)=\sum_aP_aXP_a.
\detUse in source · 7 passages
Search aliases: det
Use in source · monograph_atlas.tex, line 261
E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). It equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates. vfigrange_atlas headAn exact example of the domain restriction For the rotating line and anchor P(0), T_0(t)=cos t U(t), det T_0(t)=cos^2t. The chart fails at t=π/2, but P(π/2)=diag(0,1) is a perfectly regular rank-one proje…
Exact mathematical passage
T_0(t)=\cos t\,U(t),\qquad \det T_0(t)=\cos^2t.
\operatorname{diag}Use in source · 10 passages
Search aliases: diag operatorname
Use in source · Read passage · monograph_examples.tex, line 17
ordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have…
Exact mathematical passage
Z=\operatorname{diag}(1,-1)
\dimUse in source · 13 passages
Search aliases: dim
Use in source · monograph_atlas.tex, line 61
ment changes the local spanSCHMIDT RANK AS ACCESSIBLE MULTIPLICITYTraditional edition: Theorem II.2.2, Corollary II.2.4. Write a bipartite vector as a coefficient matrix C, with the first subsystem indexing rows. Acting only on that subsystem transforms C to AC. Every resulting row remains in the row space of C, and every independent choice of such rows can be obtained in this way. dimSpan{(A⊗ I)ψ:A∈ M_d( C)} =d rank C. vfiglocal_orbits The product vector ket00 has matrix C=diag(1,0). Its local …
Exact mathematical passage
\dim\Span\{(A\otimes I)\psi:A\in M_d(\CC)\} =d\,\rank C.
\dim_\CUse in source · 2 passages
Search aliases: dim_C dim_ C dim C dim mathbb
Use in source · Read passage · 03_supplement_03.tex, line 9
A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d× m coefficient space is equivalent to rank C=m. Consequently a cyclic seed requires m≤ d. The assumption that there is at least one row is essential …
Exact mathematical passage
\mathcal O(C)\simeq \operatorname{Row}(C)^d, \qquad \boxed{\dim_\C\mathcal O(C)=d\,\rank C.}
\distUse in source · 3 passages
Search aliases: dist operatorname
Use in source · Read passage · 02_strengthened.tex, line 1598
[Uniform self energy bounds] Let δ=dist(z,Spec D)>0, g=norm B. Then normΣ(z)≤fracg^2δ and normΣ'(z)≤fracg^2δ^2. For real E,E' in an interval at distance at least δ from Spec D, normΣ(E)-Σ(E') ≤ fracg^2δ^2absE-E'.
Exact mathematical passage
\delta=\dist(z,\Spec D)>0, \qquad g=\norm B.
\dot PUse in source · 27 passages
Search aliases: dot P dot
Use in source · monograph_atlas.tex, line 231
vpage13Differentiate the subspace itselfTHE CANONICAL PACKET CONNECTIONTraditional edition: Theorem II.6.1, Theorem II.6.2. Continue with P(t)=U(t)diag(1,0)U(t)^* and Q(t)=I-P(t). Differentiating P^2=P gives dot P P+Pdot P=dot P, Pdot P P=0. The second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement. vfigconnection For this complete pair of packets, define C=dot P P+dot Q Q. Exact calculation yields the const…
Exact mathematical passage
\dot P P+P\dot P=\dot P, \qquad P\dot P P=0.
\dot P_aUse in source · 16 passages
Search aliases: dot P_a dot
Use in source · monograph_atlas.tex, line 244
n For this complete pair of packets, define C=dot P P+dot Q Q. Exact calculation yields the constant matrix C=J, with C^*=-C and [C,P]=dot P. At t=0, the displayed small matrices allow these identities to be checked by direct multiplication. headFrom two packets to a complete family For differentiable orthogonal projectors P_a summing to the identity, the paper constructs C=∑_adot P_aP_a=frac12∑_a[dot P_a,P_a]. This single connection reproduces the motion of every packet in the family. No basis …
Exact mathematical passage
C=\sum_a\dot P_aP_a=\frac12\sum_a[\dot P_a,P_a].
\dot P_bUse in source · 9 passages
Search aliases: dot P_b dot
Use in source · Read passage · 01_finite_core.tex, line 532
[Canonical packet connection] Define C(t) = frac12∑_a[dot P_a(t),P_a(t)] = ∑_adot P_a(t)P_a(t). Then C(t)^*=-C(t) and [C(t),P_b(t)]=dot P_b(t) for every packet b.
Exact mathematical passage
\boxed{ [C(t),P_b(t)]=\dot P_b(t) }
\dot QUse in source · 1 passage
Search aliases: dot Q dot
Use in source · monograph_atlas.tex, line 238
ional edition: Theorem II.6.1, Theorem II.6.2. Continue with P(t)=U(t)diag(1,0)U(t)^* and Q(t)=I-P(t). Differentiating P^2=P gives dot P P+Pdot P=dot P, Pdot P P=0. The second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement. vfigconnection For this complete pair of packets, define C=dot P P+dot Q Q. Exact calculation yields the constant matrix C=J, with C^*=-C and [C,P]=dot P. At t=0, the displayed small matr…
Exact mathematical passage
C=\dot P P+\dot Q Q.
\dot U^*Use in source · 2 passages
Search aliases: dot U^* dot
Use in source · Read passage · 03_supplement_02.tex, line 102
The explicitly defined coordinate curve y(s)=U(s)^*ψ(s) is differentiable at t, with dot y=dot U^*ψ(t)+U(t)^*dotψ. It satisfies idot y=H_effy(t), H_eff=U(t)^*HU(t)-iU(t)^*dot U. The connection term -iU(t)^*dot U is Hermitian. If H^*=H, then H_eff^*=H_eff.
Exact mathematical passage
\dot y=\dot U^*\psi(t)+U(t)^*\dot\psi.
\dot VUse in source · 1 passage
Search aliases: dot V dot
Use in source · Read passage · 02_strengthened.tex, line 704
[Moving frame Hamiltonian] Let U(t) solve idot U(t)=H(t)U(t), U(0)=I, where H(t)=H(t)^* is continuous on the compact time interval under consideration. Set V(t)=W(t)^*U(t). Then idot V(t)=H_rm mov(t)V(t), where H_rm mov(t) = W(t)^*H(t)W(t)- i W(t)^*dot W(t) = W(t)^*(H(t)- i K(t))W(t). The moving frame Hamiltonian is self adjoint.
Exact mathematical passage
\ii\dot V(t)=H_{\rm mov}(t)V(t),
\dot WUse in source · 8 passages
Search aliases: dot W dot
Use in source · Read passage · 02_strengthened.tex, line 27
The moving packet section treats time dependent projectors. The canonical generator K(t)=∑_adot P_a(t)P_a(t) is skew adjoint and transports every packet. In the moving frame, the physical Hamiltonian acquires the geometric term - i W(t)^*dot W(t). The leakage estimate must account for the motion of the packet itself. The physical coupling QHP is therefore replaced by Q(t)(H(t)- i K(t))P(t). This is the exact finite distinction between following a moving subspace and remaining in a fixed one.
Exact mathematical passage
-\ii W(t)^*\dot W(t)
\dot W^*Use in source · 1 passage
Search aliases: dot W^* dot
Use in source · Read passage · 02_strengthened.tex, line 720
Differentiate V=W^*U. Since dot W=KW and K^*=-K, dot W^*=-W^*K. Substitution gives the equation. The first term in H_rm mov is self adjoint. The operator - i W^*dot W is self adjoint because W^*dot W is skew adjoint.
Exact mathematical passage
\dot W^*=-W^*K.
\dot xUse in source · 9 passages
Search aliases: dot x dot
Use in source · monograph_atlas.tex, line 398
vpage22The smaller system remembers the discarded oneAN EXACT QUANTUM MEMORY EQUATIONTraditional edition: Theorem II.9.8, Theorem II.9.11. Choose H=X, initial state (1,0)^T, and retain the first coordinate. The coupled equations and their solution are idot x=y, idot y=x, x(t)=cos t, y(t)=- isin t. The hidden probability returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict x(t)=1. vfigmemory headEliminate exactly, not approximately Integrat…
Exact mathematical passage
\ii\dot x=y,\qquad \ii\dot y=x,\qquad x(t)=\cos t,\quad y(t)=-\ii\sin t.
\dot yUse in source · 9 passages
Search aliases: dot y dot
Use in source · monograph_atlas.tex, line 218
1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i - i · 0, idot y=H_rm framey. This matrix is Hermitian. Omitting it would predict a constant y, contradicting the displayed coordinates. In this example the norm of the omitted time derivative is exactly one, so the failure is not a small numerical residual.
Exact mathematical passage
H_{\rm frame}=-\ii J=\mat{0&\ii\\-\ii&0}, \qquad \ii\dot y=H_{\rm frame}y.
\downarrowUse in source · 9 passages
Search aliases: downarrow
Use in source · monograph_atlas.tex, line 740
It does not assert a uniform interval width from that argument, a moving time t_N with no extra estimates, or a dimension plateau at every scale. At fixed finite N, d_N(t)→0 as tdownarrow0 and as t→∞. By contrast, first taking N→∞ at positive t and then taking tdownarrow0 yields the continuum dimension. headWhere the infinite tail is controlled At a fixed time, the proof bounds centered modes below by 16k^2 and above by 4π^2k^2. The summable Gaussian dominators control both the heat trace and it…
Exact mathematical passage
t\downarrow0
E'Use in source · 2 passages
Search aliases: E'
Use in source · Read passage · 02_strengthened.tex, line 1611
[Uniform self energy bounds] Let δ=dist(z,Spec D)>0, g=norm B. Then normΣ(z)≤fracg^2δ and normΣ'(z)≤fracg^2δ^2. For real E,E' in an interval at distance at least δ from Spec D, normΣ(E)-Σ(E') ≤ fracg^2δ^2absE-E'.
Exact mathematical passage
E,E'
e^{-(t_a+t_b)H}Use in source · 1 passage
Search aliases: e^-(t_a+t_b)H
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
\e^{-1/(4t)}Use in source · 3 passages
Search aliases: e^-1/(4t) e mathrm
Use in source · Read passage · 02_strengthened.tex, line 2125
[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.
Exact mathematical passage
d_{\rm torus}(t) = d+O\left(t^{-1}\e^{-1/(4t)}\right)
\e^{-16t\abs{k}^2}Use in source · 2 passages
Search aliases: e^-16tabsk^2 e abs mathrm abs
Use in source · Read passage · 02_strengthened.tex, line 2095
ixed integer mode, [source reference] gives λ_N(k)⟶4π^2absk^2. For centered modes, the inequality sin x≥frac2xπ, 0≤ x≤fracπ2, gives λ_N(k)≥16absk^2. Also λ_N(k)≤4π^2absk^2. Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by e^-16tabsk^2, and the moment terms by 4π^2absk^2 e^-16tabsk^2. Both dominating sequences are summable on Z^d. Dominated convergence proves the two limits. Their quotient gives spectral dimension convergence.
Exact mathematical passage
\e^{-16t\abs{k}^2},
e^{-2\ii t}Use in source · 1 passage
Search aliases: e^-2ii t e^-2 i t ii mathrm
Use in source · monograph_atlas.tex, line 680
4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phase…
Exact mathematical passage
K(t)=\frac9{25}e^{-\ii t}+\frac{16}{25}e^{-2\ii t}, \qquad K(0)=1.
e^{-2it}Use in source · 1 passage
Search aliases: e^-2it
Use in source · memory_positive.tex, line 7
weight 16/25; draw[->] (3,1.4)--(5,0.6);draw[->] (9,1.4)--(7,0.6); node[draw,align=center] at (6,0.4) K(t)=(9/25)e^-it+(16/25)e^-2it;
Exact mathematical passage
K(t)=(9/25)e^{-it}+(16/25)e^{-2it}
e^{-2\tau}Use in source · 3 passages
Search aliases: e^-2τ tau
Use in source · monograph_atlas.tex, line 422
vpage23Positive transfer is not a stochastic matrixSPECTRAL POSITIVITY IS BASIS INDEPENDENTTraditional edition: Theorem II.10.1, Theorem II.10.3. For H=I-X, the two eigenvalues are zero and two. Its Euclidean transfer is T(τ)=e^-τ H =frac12mat1+q · 1-q 1-q · 1+q, q=e^-2τ. For τ≥0, the eigenvalues 1,q lie in (0,1], and multiplying two such matrices replaces q,r by qr. This is the semigroup law in spectral coordinates. vfigeuclidean At q=1/2, the displayed T happens also to have nonnegative entrie…
Exact mathematical passage
T(\tau)=e^{-\tau H} =\frac12\mat{1+q&1-q\\1-q&1+q},\qquad q=e^{-2\tau}.
\e^{-4\pi^2t\abs{k}^2}Use in source · 3 passages
Search aliases: e^-4π^2tabsk^2 e pi abs mathrm pi abs
Use in source · Read passage · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
e^{-4\tau}Use in source · 1 passage
Search aliases: e^-4τ tau
Use in source · monograph_atlas.tex, line 442
vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESTraditional edition: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect…
Exact mathematical passage
K(\tau)=\frac{1+2e^{-2\tau}+e^{-4\tau}}4 =\frac{(1+e^{-2\tau})^2}4, \qquad d_s(\tau)=-2\tau\frac{K'(\tau)}{K(\tau)} =\frac{8\tau}{e^{2\tau}+1}.
\e^{-a(E_j-E_0)}Use in source · 1 passage
Search aliases: e^-a(E_j-E_0) e mathrm
Use in source · Read passage · 02_strengthened.tex, line 1852
Diagonalize H. Each eigenvalue of T is e^-a(E_j-E_0) and therefore lies in (0,1]. The power law follows from the functional calculus.
Exact mathematical passage
\e^{-a(E_j-E_0)}
\e^{-a(H-E_0I)}Use in source · 1 passage
Search aliases: e^-a(H-E_0I) e mathrm
Use in source · Read passage · 02_strengthened.tex, line 1838
Let H=H^* on a nonzero finite dimensional Hilbert space and let E_0=minSpec H. For a lattice time step a>0, define T= e^-a(H-E_0I).
Exact mathematical passage
T=\e^{-a(H-E_0I)}.
\e^{-\abs{n}^2/(4t)}Use in source · 2 passages
Search aliases: e^-absn^2/(4t) e abs mathrm abs
Use in source · Read passage · 02_strengthened.tex, line 2107
[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.
Exact mathematical passage
Z_d(t) = \sum_{k\in\ZZ^d}\e^{-4\pi^2t\abs{k}^2} = (4\pi t)^{-d/2} \sum_{n\in\ZZ^d}\e^{-\abs{n}^2/(4t)}.
\e^{-aH_T}Use in source · 1 passage
Search aliases: e^-aH_T e mathrm
Use in source · Read passage · 02_strengthened.tex, line 1867
[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.
Exact mathematical passage
T=\e^{-aH_T}
e^{-H_n}Use in source · 1 passage
Search aliases: e^-H_n
Use in source · monograph_atlas.tex, line 797
For transfers on two states T_n=diag(1,q_n)=e^-H_n, with H_n=diag(0,-log q_n) at unit time, the error in trace norm is 2^-(n+2) and the heat traces converge to 3/2. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat trace alone does not determine that quotient. headThe retained evidence and the remaining open targets The extension includes retained exact checks ove…
Exact mathematical passage
T_n=\diag(1,q_n)=e^{-H_n}
e^{-i(t-s)H_{QQ}}Use in source · 1 passage
Search aliases: e^-i(t-s)H_QQ
Use in source · Read passage · 01_finite_core.tex, line 973
idot y · =H_QPx+H_QQy. Solving the second equation gives y(t) =e^-itH_QQy(0) -i∫_0^t e^-i(t-s)H_QQH_QPx(s) ds. Substitution yields the exact Volterra equation idot x(t) = · H_PPx(t) +H_PQe^-itH_QQy(0)
Exact mathematical passage
y(t) =e^{-itH_{QQ}}y(0) -i\int_0^t e^{-i(t-s)H_{QQ}}H_{QP}x(s)\,ds.
\e^{-\ii sH}Use in source · 1 passage
Search aliases: e^-ii sH e^- i sH e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 2445
The summability condition makes (H_n) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives e^- i tH_n- e^- i tH =- i∫_0^t e^- i(t-s)H_n(H_n-H) e^- i sH d s. Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamiltonians are contractions.
Exact mathematical passage
\e^{-\ii tH_n}-\e^{-\ii tH} =-\ii\int_0^t \e^{-\ii(t-s)H_n}(H_n-H)\e^{-\ii sH}\dd s.
\e^{-\ii sH_{\rm d}}Use in source · 1 passage
Search aliases: e^-ii sH_rm d e^- i sH_rm d e ii rm mathrm mathrm rm
Use in source · Read passage · 02_strengthened.tex, line 582
Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.
Exact mathematical passage
Q\e^{-\ii tH}P = -\ii \int_0^t Q\e^{-\ii(t-s)H}H_{\rm o}\e^{-\ii sH_{\rm d}}P \dd s.
e^{-\ii t}Use in source · 1 passage
Search aliases: e^-ii t e^- i t ii mathrm
Use in source · monograph_atlas.tex, line 680
4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phase…
Exact mathematical passage
K(t)=\frac9{25}e^{-\ii t}+\frac{16}{25}e^{-2\ii t}, \qquad K(0)=1.
\e^{-\ii tA}Use in source · 1 passage
Search aliases: e^-ii tA e^- i tA e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 1770
If B=0, the retained and hidden equations decouple. Conversely, suppose that every solution with y(0)=0 satisfies x(t)= e^- i tAx(0). The full block equation gives ddot x(0) = -(A^2+BB^*)x(0), while the declared local equation gives ddot x(0)=-A^2x(0). Since this holds for every retained initial vector, BB^*=0. Hence B=0.
Exact mathematical passage
x(t)=\e^{-\ii tA}x(0)
\e^{-\ii tD}Use in source · 3 passages
Search aliases: e^-ii tD e^- i tD e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 1680
[Exact retained memory equation] The retained coordinate satisfies idot x(t) = · Ax(t)+B e^- i tDy(0) · - i∫_0^tK(t-s)x(s) d s, where K(t)=B e^- i tDB^*. For Res>0, ∫_0^∞ e^-stK(t) d t = B(s+ i D)^-1B^*.
Exact mathematical passage
\begin{aligned} \ii\dot x(t) ={}&Ax(t)+B\e^{-\ii tD}y(0)\\ &-\ii\int_0^tK(t-s)x(s)\dd s, \end{aligned}
e^{-\ii tD}Use in source · 1 passage
Search aliases: e^-ii tD e^- i tD ii mathrm
Use in source · monograph_atlas.tex, line 696
A nonzero initial hidden vector also supplies the forcing term Be^- i tDy(0). It is additional data, not part of the convolution kernel. The retained checks cover the displayed kernel values, exact Gram factorization, and acceleration defect. The integral equation follows from the written elimination argument. takeawayMemory is positive definite through a Gram factorization, not through pointwise signs. Exact bare local closure requires the coupling to vanish.
Exact mathematical passage
Be^{-\ii tD}y(0)
\e^{-\ii tE_r}Use in source · 1 passage
Search aliases: e^-ii tE_r e^- i tE_r e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 1744
[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.
Exact mathematical passage
K(t)=\sum_r\e^{-\ii tE_r}BQ_rB^*.
\e^{-\ii tH}Use in source · 11 passages
Search aliases: e^-ii tH e^- i tH e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 525
[Exact invariant packets] The following conditions are equivalent: item The Hamiltonian commutes with each projector: [H,P_a]=0 for every a. item Transfer blocks between distinct packets vanish: P_bHP_a=0 whenever a≠ b. item Every packet P_a H is invariant under e^- i tH for every real t. When they hold, H=∑_aP_aHP_a and e^- i tH=∑_aP_a e^- i tHP_a.
Exact mathematical passage
\e^{-\ii tH}
e^{-\ii tH}Use in source · 3 passages
Search aliases: e^-ii tH e^- i tH ii mathrm
Use in source · monograph_atlas.tex, line 31
H · Self-adjoint Hamiltonian · It generates evolution in real time through e^- i tH.
Exact mathematical passage
e^{-\ii tH}
\e^{-\ii tH_n}Use in source · 4 passages
Search aliases: e^-ii tH_n e^- i tH_n e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 2428
[Norm summable Hamiltonian tower] Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{\abs t\leq T} \norm{\e^{-\ii tH_n}-\e^{-\ii tH}} \leq T\norm{H_n-H}.
e^{-\ii tH_n}Use in source · 1 passage
Search aliases: e^-ii tH_n e^- i tH_n ii mathrm
Use in source · monograph_atlas.tex, line 779
For the norm route, work instead on the single fixed space C^2 and set H_n=diag(0,1+2^-n), H=diag(0,1). Here normH_n+1-H_n=2^-(n+1) and normH_n-H=2^-n. The summable tail supplies an error bound for every n. For |t|≤ T, the paper's Duhamel estimate gives norme^- i tH_n-e^- i tH≤ T2^-n. The Euclidean counterpart holds on 0≤ t≤ T because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals. headExecution scope Ex…
Exact mathematical passage
\norm{e^{-\ii tH_n}-e^{-\ii tH}}\leq T2^{-n}.
\e^{-\ii tH_\ell}Use in source · 1 passage
Search aliases: e^-ii tH_ℓ e^- i tH_ℓ e ii ell mathrm mathrm ell
Use in source · Read passage · 02_strengthened.tex, line 2380
[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\e^{-\ii tH_\infty}\iota_\ell = \iota_\ell\e^{-\ii tH_\ell}
\e^{-\ii tH_\infty}Use in source · 1 passage
Search aliases: e^-ii tH_∞ e^- i tH_∞ e ii infty mathrm mathrm infty
Use in source · Read passage · 02_strengthened.tex, line 2380
[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\e^{-\ii tH_\infty}\iota_\ell = \iota_\ell\e^{-\ii tH_\ell}
\e^{-\ii(t-s)D}Use in source · 1 passage
Search aliases: e^-ii(t-s)D e^- i(t-s)D e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 1705
The hidden equation is idot y=B^*x+Dy. Variation of constants gives y(t)= e^- i tDy(0) - i∫_0^t e^- i(t-s)DB^*x(s) d s. Substitute into idot x=Ax+By. The Laplace formula follows by diagonalizing D or integrating the norm convergent finite dimensional exponential.
Exact mathematical passage
y(t)=\e^{-\ii tD}y(0) -\ii\int_0^t\e^{-\ii(t-s)D}B^*x(s)\dd s.
\e^{-\ii(t-s)H}Use in source · 1 passage
Search aliases: e^-ii(t-s)H e^- i(t-s)H e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 582
Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.
Exact mathematical passage
Q\e^{-\ii tH}P = -\ii \int_0^t Q\e^{-\ii(t-s)H}H_{\rm o}\e^{-\ii sH_{\rm d}}P \dd s.
\e^{-\ii(t-s)H_n}Use in source · 1 passage
Search aliases: e^-ii(t-s)H_n e^- i(t-s)H_n e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 2445
The summability condition makes (H_n) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives e^- i tH_n- e^- i tH =- i∫_0^t e^- i(t-s)H_n(H_n-H) e^- i sH d s. Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamiltonians are contractions.
Exact mathematical passage
\e^{-\ii tH_n}-\e^{-\ii tH} =-\ii\int_0^t \e^{-\ii(t-s)H_n}(H_n-H)\e^{-\ii sH}\dd s.
\e^{-\ii(t_j-t_k)D}Use in source · 1 passage
Search aliases: e^-ii(t_j-t_k)D e^- i(t_j-t_k)D e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 1727
Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.
Exact mathematical passage
\norm w^2 = \sum_{j,k} \ip{B^*v_j}{\e^{-\ii(t_j-t_k)D}B^*v_k} = \sum_{j,k} \ip{v_j}{K(t_j-t_k)v_k}.
e^{-\ii\pi L/2}Use in source · 2 passages
Search aliases: e^-iiπ L/2 e^- iπ L/2 ii pi mathrm pi
Use in source · monograph_atlas.tex, line 495
lculation of transfer to the opposite siteSEEDS, CHARACTERS, INTERFERENCE, AND DEPHASINGTraditional edition: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4. Return to the shift on four sites S and Laplacian L=2I-S-S^*. The same character projectors have energies (0,2,4,2). At time t=π/2, their evolution phases are (1,-1,1,-1), exactly the character eigenvalues of S^2. Hence e^- iπ L/2=S^2, ket0⟼ket2. vfigperfect_transfer At halfway time π/4, the actual amplitude vector is (- i,1, i,1)^T/2, giv…
Exact mathematical passage
e^{-\ii\pi L/2}=S^2,\qquad \ket0\longmapsto\ket2.
e^{-it}Use in source · 1 passage
Search aliases: e^-it
Use in source · memory_positive.tex, line 7
weight 16/25; draw[->] (3,1.4)--(5,0.6);draw[->] (9,1.4)--(7,0.6); node[draw,align=center] at (6,0.4) K(t)=(9/25)e^-it+(16/25)e^-2it;
Exact mathematical passage
K(t)=(9/25)e^{-it}+(16/25)e^{-2it}
e^{-itH_{QQ}}Use in source · 3 passages
Search aliases: e^-itH_QQ
Use in source · Read passage · 01_finite_core.tex, line 973
idot y · =H_QPx+H_QQy. Solving the second equation gives y(t) =e^-itH_QQy(0) -i∫_0^t e^-i(t-s)H_QQH_QPx(s) ds. Substitution yields the exact Volterra equation idot x(t) = · H_PPx(t) +H_PQe^-itH_QQy(0)
Exact mathematical passage
y(t) =e^{-itH_{QQ}}y(0) -i\int_0^t e^{-i(t-s)H_{QQ}}H_{QP}x(s)\,ds.
\e^{-na(E_j-E_0)}Use in source · 1 passage
Search aliases: e^-na(E_j-E_0) e mathrm
Use in source · Read passage · 02_strengthened.tex, line 1934
In the spectral decomposition, T^n-P_0 has eigenvalues zero on the ground space and e^-na(E_j-E_0) on the excited eigenspaces. The largest is e^-naΔ.
Exact mathematical passage
\e^{-na(E_j-E_0)}
\e^{-na(H-E_0I)}Use in source · 1 passage
Search aliases: e^-na(H-E_0I) e mathrm
Use in source · Read passage · 02_strengthened.tex, line 1844
[Finite Euclidean transfer] The operator T is positive definite, self adjoint, and contractive. Its spectrum lies in (0,1], and T^n= e^-na(H-E_0I) for every n∈ N.
Exact mathematical passage
T^n=\e^{-na(H-E_0I)}
\e^{-naH_T}Use in source · 1 passage
Search aliases: e^-naH_T e mathrm
Use in source · Read passage · 02_strengthened.tex, line 1864
[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.
Exact mathematical passage
T^n=\e^{-naH_T}.
\e^{-na\Delta}Use in source · 2 passages
Search aliases: e^-naΔ e Delta mathrm Delta
Use in source · Read passage · 02_strengthened.tex, line 1928
[Ground packet convergence] Let P_0 be the spectral projector of H at E_0, and let Δ=min{E_j-E_0:E_j>E_0}. If an excited eigenvalue exists, then normT^n-P_0= e^-naΔ.
Exact mathematical passage
\norm{T^n-P_0}=\e^{-na\Delta}.
\e^{-st}Use in source · 1 passage
Search aliases: e^-st e mathrm
Use in source · Read passage · 02_strengthened.tex, line 1692
· - i∫_0^tK(t-s)x(s) d s, where K(t)=B e^- i tDB^*. For Res>0, ∫_0^∞ e^-stK(t) d t = B(s+ i D)^-1B^*.
Exact mathematical passage
\int_0^\infty\e^{-st}K(t)\dd t = B(s+\ii D)^{-1}B^*.
e^{-t(H-E_0I)}Use in source · 1 passage
Search aliases: e^-t(H-E_0I)
Use in source · monograph_atlas.tex, line 714
For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = …
Exact mathematical passage
e^{-t(H-E_0I)}
e^{-t(x-e)}Use in source · 2 passages
Search aliases: e^-t(x-e)
Use in source · Read passage · 03_supplement_03.tex, line 143
Connecting power series and spectral calculus. The shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator that was actually de…
Exact mathematical passage
\boxed{T_e(t)=f_t(H),\qquad f_t(x)=e^{-t(x-e)}.}
e^{-t_aH}Use in source · 1 passage
Search aliases: e^-t_aH
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
\e^{-tH}Use in source · 1 passage
Search aliases: e^-tH e mathrm
Use in source · Read passage · 02_strengthened.tex, line 2435
Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{\e^{-tH_n}-\e^{-tH}} \leq T\norm{H_n-H}.
\e^{-tH_n}Use in source · 1 passage
Search aliases: e^-tH_n e mathrm
Use in source · Read passage · 02_strengthened.tex, line 2435
Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{\e^{-tH_n}-\e^{-tH}} \leq T\norm{H_n-H}.
\e^{-tH_\ell}Use in source · 1 passage
Search aliases: e^-tH_ℓ e ell mathrm ell
Use in source · Read passage · 02_strengthened.tex, line 2386
imit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\e^{-tH_\infty}\iota_\ell = \iota_\ell\e^{-tH_\ell}
\e^{-tH_\infty}Use in source · 1 passage
Search aliases: e^-tH_∞ e infty mathrm infty
Use in source · Read passage · 02_strengthened.tex, line 2386
imit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\e^{-tH_\infty}\iota_\ell = \iota_\ell\e^{-tH_\ell}
\e^{-tL_n}Use in source · 3 passages
Search aliases: e^-tL_n e mathrm
Use in source · Read passage · 02_strengthened.tex, line 2512
[Heat trace convergence from trace norm] Let L_n,L be nonnegative self adjoint operators on one Hilbert space. For a fixed t>0, assume their heat operators are trace class and that e^-tL_n⟶ e^-tL in trace norm. Then Tr( e^-tL_n)⟶Tr( e^-tL). If also the heat moment operators are trace class and L_n e^-tL_n⟶ L e^-tL in trace norm and the limiting heat trace is nonzero, then the spectral dimensions defined by the heat moment ratio converge at that time.
Exact mathematical passage
\e^{-tL_n}\longrightarrow\e^{-tL}
e^{-t\lambda}Use in source · 1 passage
Search aliases: e^-tλ lambda
Use in source · monograph_atlas.tex, line 728
At the single time t=log2/16, two small grids admit exact rational heat weights: c|c|c|c|c N · spectrum · Z_N · ∑λ e^-tλ · d_N/log2 2 · (0,16) · 3/2 · 8 · 2/3 4 · (0,32,32,64) · 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less tha…
Exact mathematical passage
\begin{array}{c|c|c|c|c} N&\text{spectrum}&Z_N&\sum\lambda e^{-t\lambda}&d_N/\log2\\\hline 2&(0,16)&3/2&8&2/3\\ 4&(0,32,32,64)&25/16&20&8/5 \end{array}
\e^{-t\lambda_N(k)}Use in source · 2 passages
Search aliases: e^-tλ_N(k) e lambda mathrm lambda
Use in source · Read passage · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
e^{-t\lambda_N(k)}Use in source · 2 passages
Search aliases: e^-tλ_N(k) lambda
Use in source · monograph_atlas.tex, line 725
vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSTraditional edition: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).
Exact mathematical passage
Z_N(t)=\sum_ke^{-t\lambda_N(k)}
e^{-\tau H}Use in source · 1 passage
Search aliases: e^-τ H tau
Use in source · monograph_atlas.tex, line 422
vpage23Positive transfer is not a stochastic matrixSPECTRAL POSITIVITY IS BASIS INDEPENDENTTraditional edition: Theorem II.10.1, Theorem II.10.3. For H=I-X, the two eigenvalues are zero and two. Its Euclidean transfer is T(τ)=e^-τ H =frac12mat1+q · 1-q 1-q · 1+q, q=e^-2τ. For τ≥0, the eigenvalues 1,q lie in (0,1], and multiplying two such matrices replaces q,r by qr. This is the semigroup law in spectral coordinates. vfigeuclidean At q=1/2, the displayed T happens also to have nonnegative entrie…
Exact mathematical passage
T(\tau)=e^{-\tau H} =\frac12\mat{1+q&1-q\\1-q&1+q},\qquad q=e^{-2\tau}.
e^{-\tau(E_j-E_0)}Use in source · 1 passage
Search aliases: e^-τ(E_j-E_0) tau
Use in source · Read passage · 01_finite_core.tex, line 1072
Diagonalize H=∑_jE_jP_j. Then T(τ)=∑_j e^-τ(E_j-E_0)P_j. Every eigenvalue is strictly positive and at most one. Multiplication adds the exponents, proving the semigroup law and T(0)=I. Positivity here is operator positivity, not an assertion that every matrix entry is nonnegative.
Exact mathematical passage
T(\tau)=\sum_j e^{-\tau(E_j-E_0)}P_j.
e^{-\tau(H-E_0I)}Use in source · 1 passage
Search aliases: e^-τ(H-E_0I) tau
Use in source · Read passage · 01_finite_core.tex, line 1056
[Finite Euclidean transfer] Let H be Hermitian on a nonzero finite-dimensional Hilbert space. For all real τ,σ≥0, T(τ)=e^-τ(H-E_0I), E_0=minSpec H, is positive definite and satisfies T(τ+σ)=T(τ)T(σ). Its spectrum is contained in (0,1].
Exact mathematical passage
T(\tau)=e^{-\tau(H-E_0I)}, \qquad E_0=\min\Spec H,
e^{-\tau\lambda_j}Use in source · 2 passages
Search aliases: e^-τλ_j tau lambda
Use in source · Read passage · 01_finite_core.tex, line 1116
Let L≥0 be a finite Laplacian on N≥1 states, with eigenvalues λ_j and at least one zero eigenvalue. For diffusion time τ>0, define the normalized heat trace K(τ)=frac1N∑_j e^-τλ_j and the scale-dependent spectral dimension d_s(τ) =-2fracdlog K(τ)dlogτ. For a finite system, d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differentiation gives d_s(τ)=2τ frac∑_jλ_j e^-τλ_j∑_j e^-τλ_j. It tends to zero as τdownarrow0 and, under the stated assumption of a zero mode, as τ→∞. If a g…
Exact mathematical passage
K(\tau)=\frac1N\sum_j e^{-\tau\lambda_j}
e^{2\pi i/N}Use in source · 1 passage
Search aliases: e^2π i/N pi
Use in source · Read passage · 01_finite_core.tex, line 192
Let N≥1 be an integer, and let S be a unitary symmetry whose order divides N: S^N=I. Set ω=e^2π i/N.
Exact mathematical passage
\omega=e^{2\pi i/N}
\e^{2\pi\ii/N}Use in source · 1 passage
Search aliases: e^2πii/N e^2π i/N e pi ii mathrm pi mathrm
Use in source · Read passage · 02_strengthened.tex, line 270
Let N≥1 be an integer, S unitary with S^N=I, and ω= e^2π i/N.
Exact mathematical passage
\omega=\e^{2\pi\ii/N}
e^{2\tau}Use in source · 1 passage
Search aliases: e^2τ tau
Use in source · monograph_atlas.tex, line 442
vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESTraditional edition: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect…
Exact mathematical passage
K(\tau)=\frac{1+2e^{-2\tau}+e^{-4\tau}}4 =\frac{(1+e^{-2\tau})^2}4, \qquad d_s(\tau)=-2\tau\frac{K'(\tau)}{K(\tau)} =\frac{8\tau}{e^{2\tau}+1}.
\e^{\ii t_kD}Use in source · 1 passage
Search aliases: e^ii t_kD e^ i t_kD e ii mathrm mathrm
Use in source · Read passage · 02_strengthened.tex, line 1723
Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.
Exact mathematical passage
w=\sum_{k=1}^{m}\e^{\ii t_kD}B^*v_k.
e^{\ii t_kD}Use in source · 1 passage
Search aliases: e^ii t_kD e^ i t_kD ii mathrm
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
e^{\ii\phi}Use in source · 1 passage
Search aliases: e^iiϕ e^ iϕ ii phi mathrm phi
Use in source · monograph_atlas.tex, line 283
n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ…
Exact mathematical passage
u_N=(\cos(\theta/2),e^{\ii\phi}\sin(\theta/2))^T
e^{isH}Use in source · 2 passages
Search aliases: e^isH
Use in source · Read passage · 01_finite_core.tex, line 442
Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives the desired inequality. This argument includes positive, negative and zero time.
Exact mathematical passage
F(s)=e^{isH}Pe^{-isH}
e^{itH}Use in source · 2 passages
Search aliases: e^itH
Use in source · Read passage · 01_finite_core.tex, line 450
Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives the desired inequality. This argument includes positive, negative and zero time.
Exact mathematical passage
e^{itH}Qe^{-itH}P=(P-F(t))P
\e^{t\Gcal}Use in source · 2 passages
Search aliases: e^tGcal e^t G e Gcal mathrm mathcal
Use in source · Read passage · 02_strengthened.tex, line 1244
a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.
Exact mathematical passage
\e^{t\Gcal}
e^{zH}Use in source · 2 passages
Search aliases: e^zH
Use in source · Read passage · 03_supplement_01.tex, line 59
Symmetry and measurement. Exact symmetry: source Theorem 5.1. If HP_a=P_aH for every packet, then P_bHP_a=0 (b≠ a), H=∑_aP_aHP_a. In a complex Banach algebra, commutation also propagates to the actual exponential: e^zHP_a=P_ae^zH (z∈ C). Consequently, e^zH=∑_aP_ae^zHP_a, P_be^zHP_a=0 (b≠ a). Taking z=-it gives evolution in real time. The Lean proof applies the exponential commutation theorem directly, so invariance of the packets is a proved consequence of commutation. Self-adjointness of H is n…
Exact mathematical passage
e^{zH}P_a=P_ae^{zH}\qquad(z\in\C).
E_0Use in source · 17 passages
Search aliases: E_0
Use in source · monograph_atlas.tex, line 714
For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = …
Exact mathematical passage
E_0\ne0
E_0^*Use in source · 1 passage
Search aliases: E_0^*
Use in source · Read passage · 03_supplement_05.tex, line 211
A genuine channel can still be uncorrectable. For full qubit dephasing, E_0=diag(1,0) and E_1=diag(0,1), the Kraus normalization is checked. But E_0^*E_0 cannot be a scalar multiple of I, as its two diagonal entries differ. Necessity therefore proves that no finite recovery channel corrects the full code space. The case of the zero code is also checked independently.
Exact mathematical passage
E_0^*E_0
E_1Use in source · 5 passages
Search aliases: E_1
Use in source · monograph_atlas.tex, line 597
vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATATraditional edition: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0
Exact mathematical passage
E_1=(3/5)P
E_{12}Use in source · 2 passages
Search aliases: E_12
Use in source · Read passage · 01_finite_core.tex, line 1217
ty gives A_ABΠ^phys=0, Π^phys A_ABΠ^phys=0 automatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, C_1=diag(1,0,0) and C_2=E_12+E_21 annihilate the same third coordinate, while their commutator E_12-E_21 is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression v…
Exact mathematical passage
C_2=E_{12}+E_{21}
E_2Use in source · 1 passage
Search aliases: E_2
Use in source · monograph_atlas.tex, line 597
vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATATraditional edition: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0
Exact mathematical passage
E_2=(4/5)P
E_{21}Use in source · 2 passages
Search aliases: E_21
Use in source · Read passage · 01_finite_core.tex, line 1217
ty gives A_ABΠ^phys=0, Π^phys A_ABΠ^phys=0 automatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, C_1=diag(1,0,0) and C_2=E_12+E_21 annihilate the same third coordinate, while their commutator E_12-E_21 is nonzero. With zero proposed structure coefficients, the anomaly is nonzero but its physical compression v…
Exact mathematical passage
C_2=E_{12}+E_{21}
E_3Use in source · 1 passage
Search aliases: E_3
Use in source · monograph_atlas.tex, line 597
vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATATraditional edition: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0
Exact mathematical passage
E_3=0
E_{ab}Use in source · 1 passage
Search aliases: E_ab
Use in source · monograph_atlas.tex, line 70
For the exact calculation, apply the four matrix units E_ab to each seed matrix, flatten the outputs, and compute their column rank. The observed dimensions are two and four, respectively. The calculation directly exhibits the independent rows counted by the theorem. takeawaySchmidt rank measures the remote multiplicity directions accessible to the local operator span. It is not permission for unrestricted deterministic local state preparation.
Exact mathematical passage
E_{ab}
E_b^*Use in source · 1 passage
Search aliases: E_b^*
Use in source · Read passage · 03_supplement_05.tex, line 169
Cross terms for the original errors. For arbitrary positive Gram data, the same recovery satisfies R(E_aρ E_b^*)=C_baρ (Pρ P=ρ). The index order ba is intentional: the defining Gram relation is PE_a^*E_bP=C_abP. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to identify the original Gram entries.
Exact mathematical passage
\boxed{\mathcal R(E_a\rho E_b^*)=C_{ba}\rho\qquad(P\rho P=\rho).}
E_{ii}^{\lambda}Use in source · 1 passage
Search aliases: E_ii^λ lambda
Use in source · Read passage · 02_strengthened.tex, line 259
x an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
P_\lambda=\sum_{i=1}^{d_\lambda}E_{ii}^{\lambda}.
E_{ij}^{\lambda}Use in source · 2 passages
Search aliases: E_ij^λ lambda
Use in source · Read passage · 02_strengthened.tex, line 243
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
E_{ij}^{\lambda} = \frac{d_\lambda}{\abs{G}} \sum_{g\in G} \bigl[\pi_\lambda(g^{-1})\bigr]_{ji}U(g).
E_{i\ell}^{\lambda}Use in source · 1 passage
Search aliases: E_iℓ^λ ell lambda
Use in source · Read passage · 02_strengthened.tex, line 251
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
E_{ij}^{\lambda}E_{k\ell}^{\mu} = \delta_{\lambda\mu}\delta_{jk}E_{i\ell}^{\lambda}, \qquad \bigl(E_{ij}^{\lambda}\bigr)^*=E_{ji}^{\lambda},
E_jUse in source · 6 passages
Search aliases: E_j
Use in source · Read passage · monograph_examples.tex, line 17
resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ. [H]centering includegraphics[width=.94textwidth]m…
Exact mathematical passage
\sum_j E_j^*E_j=4I
e_jUse in source · 2 passages
Search aliases: e_j
Use in source · Read passage · 03_supplement_02.tex, line 17
0 · x≠ a. The seed is ψ_0=ket0. Define a translation operator by (T_a v)(x)=v(x-a). At position j, the local Pauli X is X_j=T_e_j, where e_j has a single nonzero binary coordinate. The formal circuit uses the identity when a_j=0 and X_j when a_j=1.
Exact mathematical passage
X_j=T_{e_j}
E_j^*Use in source · 1 passage
Search aliases: E_j^*
Use in source · Read passage · monograph_examples.tex, line 17
resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ. [H]centering includegraphics[width=.94textwidth]m…
Exact mathematical passage
\sum_j E_j^*E_j=4I
E_{ji}^{\lambda}Use in source · 1 passage
Search aliases: E_ji^λ lambda
Use in source · Read passage · 02_strengthened.tex, line 251
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
E_{ij}^{\lambda}E_{k\ell}^{\mu} = \delta_{\lambda\mu}\delta_{jk}E_{i\ell}^{\lambda}, \qquad \bigl(E_{ij}^{\lambda}\bigr)^*=E_{ji}^{\lambda},
E_kUse in source · 13 passages
Search aliases: E_k
Use in source · monograph_atlas.tex, line 29
E_k · Positive measurement effect · It gives the outcome probability Tr(E_kρ).
Exact mathematical passage
E_k
E_{k\ell}^{\mu}Use in source · 1 passage
Search aliases: E_kℓ^μ ell mu
Use in source · Read passage · 02_strengthened.tex, line 251
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
E_{ij}^{\lambda}E_{k\ell}^{\mu} = \delta_{\lambda\mu}\delta_{jk}E_{i\ell}^{\lambda}, \qquad \bigl(E_{ij}^{\lambda}\bigr)^*=E_{ji}^{\lambda},
E_rUse in source · 4 passages
Search aliases: E_r
Use in source · Read passage · 01_finite_core.tex, line 792
Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environment can distinguish logical states and exact correction is impossible.
Exact mathematical passage
E_1,\ldots,E_r
E_xUse in source · 1 passage
Search aliases: E_x
Use in source · monograph_atlas.tex, line 253
vpage14A chart failure is not a singular fiberTHE RANGE BUNDLE IS CONSTRUCTED LOCALLYTraditional edition: Theorem II.6.7. Details of the range atlas: saved v0.6.0 supplement, pp. 2--4. The fiber of a projector field is the actual fixed subspace E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). It equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides loca…
Exact mathematical passage
E_x=\{v:P(x)v=v\}
E_zUse in source · 2 passages
Search aliases: E_z
Use in source · Read passage · 03_supplement_04.tex, line 130
very operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent superpositions. This construction supplies the recove…
Exact mathematical passage
E_z=\sum_a z_aV_a
E_z^*Use in source · 1 passage
Search aliases: E_z^*
Use in source · Read passage · 03_supplement_04.tex, line 131
identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent superpositions. This construction supplies the recovery stage used in the full Knill--Laflamme proof that foll…
Exact mathematical passage
\mathcal R(E_z\rho E_z^*)=\left(\sum_a z_a\overline{z_a}\right)\rho.
E_\ellUse in source · 6 passages
Search aliases: E_ℓ ell
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
E_\ell^*Use in source · 1 passage
Search aliases: E_ℓ^* ell
Use in source · Read passage · 01_finite_core.tex, line 1019
order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
C_\ell=E_\ell^*
\mathrm{eff}Use in source · 7 passages
Search aliases: eff mathrm
Use in source · Read passage · 01_finite_core.tex, line 907
[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the appropriately sized scalar matrices minus the indicated blocks.
Exact mathematical passage
\boxed{ H_{\mathrm{eff}}(z) = H_{PP} +H_{PQ}(z-H_{QQ})^{-1}H_{QP}. }
\operatorname{End}Use in source · 1 passage
Search aliases: End operatorname
Use in source · Read passage · 01_finite_core.tex, line 573
[Projected connection] Let M be a smooth finite-dimensional real manifold without boundary, let V be a finite-dimensional real normed vector space, and let P:M→End_ R(V) be smooth with P(x)^2=P(x). In the quantum application, V is the underlying real space of a finite complex Hilbert space and P is an orthogonal complex-linear projector.
Exact mathematical passage
P:M\to\operatorname{End}_{\bbR}(V)
\operatorname{End}_{\bbR}Use in source · 1 passage
Search aliases: End_bbR End_ R operatorname bbR operatorname mathbb
Use in source · Read passage · 01_finite_core.tex, line 573
[Projected connection] Let M be a smooth finite-dimensional real manifold without boundary, let V be a finite-dimensional real normed vector space, and let P:M→End_ R(V) be smooth with P(x)^2=P(x). In the quantum application, V is the underlying real space of a finite complex Hilbert space and P is an orthogonal complex-linear projector.
Exact mathematical passage
P:M\to\operatorname{End}_{\bbR}(V)
\mathrm{even}Use in source · 3 passages
Search aliases: even mathrm
Use in source · Read passage · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
\expUse in source · 9 passages
Search aliases: exp
Use in source · Read passage · 03_supplement_03.tex, line 93
Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N is proved from S^N=I, giving ρ(r+s)=ρ(r)ρ(s) and ρ(0)=I. The final matrix theorem derives the representation law …
Exact mathematical passage
\omega=\exp(2\pi i/N),\qquad \chi(r)=\omega^{\operatorname{val}(r)}.
\mathrm{extra}Use in source · 2 passages
Search aliases: extra mathrm
Use in source · Read passage · 03_supplement_04.tex, line 109
The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family I_J⊗ R_a, including the complement term. Positivity is …
Exact mathematical passage
R_{\mathrm{extra}}=Q
F'Use in source · 2 passages
Search aliases: F'
Use in source · Read passage · 01_finite_core.tex, line 443
Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives the desired inequality. This argument includes positive, negative and zero time.
Exact mathematical passage
F'(s)=i\,e^{isH}[H,P]e^{-isH},\qquad F(0)=P.
F^*Use in source · 1 passage
Search aliases: F^*
Use in source · Read passage · 02_strengthened.tex, line 1295
Exact correction of an error span means that one completely positive trace preserving recovery R satisfies R(Fρ F^*)=q_Fρ for every F in that span and every code-supported matrix ρ, with a nonnegative scalar q_F depending on F but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restores the code input with total scalar one. Individual unnormalized errors are not required to have unit recovery weight.
Exact mathematical passage
\Rcal(F\rho F^*)=q_F\rho
F_1Use in source · 1 passage
Search aliases: F_1
Use in source · monograph_atlas.tex, line 603
0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divid…
Exact mathematical passage
W=\mat{3/5&4/5&0\\-4/5&3/5&0\\0&0&1}, \quad WcW^*=\diag(1,0,0), \quad(F_1,F_2,F_3)=(P,0,0).
F_2Use in source · 1 passage
Search aliases: F_2
Use in source · monograph_atlas.tex, line 603
0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divid…
Exact mathematical passage
W=\mat{3/5&4/5&0\\-4/5&3/5&0\\0&0&1}, \quad WcW^*=\diag(1,0,0), \quad(F_1,F_2,F_3)=(P,0,0).
F_3Use in source · 1 passage
Search aliases: F_3
Use in source · monograph_atlas.tex, line 603
0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divid…
Exact mathematical passage
W=\mat{3/5&4/5&0\\-4/5&3/5&0\\0&0&1}, \quad WcW^*=\diag(1,0,0), \quad(F_1,F_2,F_3)=(P,0,0).
F_aUse in source · 3 passages
Search aliases: F_a
Use in source · Read passage · 03_supplement_05.tex, line 42
From Gram data to an actual total recovery. Assume Csucceq0 and PE_a^*E_bP=C_abP. First restrict the error operators to the code, F_a=E_aP. The finite Hermitian spectral theorem supplies a unitary U such that U^*CU=diag(λ_r), λ_r≥0.
Exact mathematical passage
F_a=E_aP
F_a^*Use in source · 1 passage
Search aliases: F_a^*
Use in source · Read passage · 03_supplement_05.tex, line 53
Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.
Exact mathematical passage
B_rP=B_r,\qquad B_r^*B_s=\delta_{rs}\lambda_rP, \qquad \sum_rB_r\rho B_r^*=\sum_aF_a\rho F_a^*.
f_{AB}^{\ \ C}Use in source · 2 passages
Search aliases: f_AB^ C
Use in source · Read passage · 01_finite_core.tex, line 1198
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
f_{AB}^{\ \ C}
f_cUse in source · 3 passages
Search aliases: f_c
Use in source · Read passage · 03_supplement_01.tex, line 188
A constraint diagnostic that vanishes automatically. Section 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives C_aP=0 for every a. Absorb the factor i and any structure coefficient into f_c, and write A_ab=C_aC_b-C_bC_a-∑_cf_cC_c. [Exact common-kernel compression] If C_cP=0 for every c, then A_abP=0, P A_abP=0. No idempotence or adjoint property of P is needed for this implication. Associativity gives A_abP=C_a(C_bP)-C_b(C_aP)-∑_c…
Exact mathematical passage
f_c
F_jUse in source · 7 passages
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Use in source · Read passage · 02_strengthened.tex, line 1323
[Orthogonal syndrome construction] Assume the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
F_j=\sum_a\overline{W_{ja}}E_a.
F_j^*Use in source · 4 passages
Search aliases: F_j^*
Use in source · Read passage · 02_strengthened.tex, line 1343
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_j^*F_\ell P = \lambda_j\delta_{j\ell}P.
F_{P\circ f}Use in source · 1 passage
Search aliases: F_P∘ f circ
Use in source · Read passage · 03_supplement_04.tex, line 86
Tensor properties and the remaining global boundary. The coefficient is proved bilinear, alternating, and supported on the range on both sides. The chain rule proves the pullback identity F_P∘ f(x)(v,w)=F_P(f(x))(Df(x)v,Df(x)w). The main theorem has global regularity hypotheses on normed parameter spaces. The pullback identity alone does not supply localization, smooth local trivializations, or gluing. The final supplement constructs the global range bundle and proves that the local connections …
Exact mathematical passage
F_{P\circ f}(x)(v,w)=F_P(f(x))(Df(x)v,Df(x)w).
F_rUse in source · 6 passages
Search aliases: F_r
Use in source · Read passage · 01_finite_core.tex, line 843
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
F_r=\sum_aU_{ar}E_aP.
F_r^*Use in source · 2 passages
Search aliases: F_r^*
Use in source · Read passage · 01_finite_core.tex, line 846
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
F_r^*F_s=\lambda_r\delta_{rs}P
F_sUse in source · 1 passage
Search aliases: F_s
Use in source · Read passage · 01_finite_core.tex, line 846
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
F_r^*F_s=\lambda_r\delta_{rs}P
f_tUse in source · 1 passage
Search aliases: f_t
Use in source · Read passage · 03_supplement_03.tex, line 143
Connecting power series and spectral calculus. The shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator that was actually de…
Exact mathematical passage
\boxed{T_e(t)=f_t(H),\qquad f_t(x)=e^{-t(x-e)}.}
F_zUse in source · 2 passages
Search aliases: F_z
Use in source · Read passage · 02_strengthened.tex, line 1338
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
F_z=\sum_az_aE_a
F_z^*Use in source · 1 passage
Search aliases: F_z^*
Use in source · Read passage · 02_strengthened.tex, line 1339
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_z^*F_zP=(z^*cz)P\geq0,
F_{\theta\phi}Use in source · 2 passages
Search aliases: F_θϕ theta phi
Use in source · monograph_atlas.tex, line 277
LOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T.…
Exact mathematical passage
F_{\theta\phi}=-\ii\Tr\!\left(P[\partial_\theta P,\partial_\phi P]\right) =\frac12\sin\theta.
F_\ellUse in source · 1 passage
Search aliases: F_ℓ ell
Use in source · Read passage · 02_strengthened.tex, line 1343
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_j^*F_\ell P = \lambda_j\delta_{j\ell}P.
\FcalUse in source · 5 passages
Search aliases: Fcal F mathcal
Use in source · Read passage · 02_strengthened.tex, line 869
[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.
Exact mathematical passage
\Fcal=\dd\Acal+\Acal\wedge\Acal
\Fcal_{\theta\phi}Use in source · 3 passages
Search aliases: Fcal_θϕ F_θϕ Fcal theta phi mathcal theta phi
Use in source · Read passage · 02_strengthened.tex, line 901
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
\Fcal_{\theta\phi} =-\ii\Tr\bigl(P[\partial_\theta P,\partial_\phi P]\bigr),
\mathrm{fine}Use in source · 1 passage
Search aliases: fine mathrm
Use in source · Read passage · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
G^*Use in source · 1 passage
Search aliases: G^*
Use in source · Read passage · 02_strengthened.tex, line 863
[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.
Exact mathematical passage
\Acal' = G^*\Acal G+G^*\dd G.
g^{-1}Use in source · 3 passages
Search aliases: g^-1
Use in source · monograph_atlas.tex, line 534
, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and three-cycle classes are c|rrr|r · e · (12) · (123) · d_λ trivial · 1 · 1 · 1 · 1 sign · 1 · -1 · 1 · 1 standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3…
Exact mathematical passage
\begin{array}{c|rrr|r} & e & (12)&(123)&d_\lambda\\\hline \text{trivial}&1&1&1&1\\ \text{sign}&1&-1&1&1\\ \text{standard}&2&0&-1&2 \end{array} \qquad P_\lambda=\frac{d_\lambda}{6}\sum_{g\in S_3}\chi_\lambda(g^{-1})U(g).
g^2Use in source · 5 passages
Search aliases: g^2
Use in source · monograph_atlas.tex, line 661
The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. Then |Σ-widetildeΣ|=|1-frac8156|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the diff…
Exact mathematical passage
|\Sigma-\widetilde\Sigma|=\left|1-\frac{81}{56}\right|=\frac{25}{56}, \qquad \frac{2g}{\delta}|B-\widetilde B| +\frac{g^2}{\delta^2}|D-\widetilde D|=\frac{21}{32}.
g_{aa}Use in source · 1 passage
Search aliases: g_aa
Use in source · Read passage · 03_supplement_06.tex, line 50
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{aa}=I,\qquad g_{bc}g_{ab}=g_{ac}.
g_{ab}Use in source · 3 passages
Search aliases: g_ab
Use in source · Read passage · 03_supplement_06.tex, line 49
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{ab}(x)=A_b(x)B_a(x)
g_{ac}Use in source · 1 passage
Search aliases: g_ac
Use in source · Read passage · 03_supplement_06.tex, line 50
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{aa}=I,\qquad g_{bc}g_{ab}=g_{ac}.
g_{bc}Use in source · 1 passage
Search aliases: g_bc
Use in source · Read passage · 03_supplement_06.tex, line 50
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{aa}=I,\qquad g_{bc}g_{ab}=g_{ac}.
G_{jk}Use in source · 1 passage
Search aliases: G_jk
Use in source · Read passage · 02_strengthened.tex, line 1876
[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.
Exact mathematical passage
G_{jk}=\ip{v_j}{T^{n_j+n_k}v_k}
\GcalUse in source · 6 passages
Search aliases: Gcal G mathcal
Use in source · Read passage · 02_strengthened.tex, line 1214
[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace preserving semigroup [source reference].
Exact mathematical passage
\Gcal(X) = -\ii[H,X] + \sum_r \left( L_rXL_r^* - \frac12L_r^*L_rX - \frac12XL_r^*L_r \right),
hUse in source · 44 passages
Search aliases: h
Use in source · monograph_atlas.tex, line 461
vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters because the displayed expression divides by h^2. vfigdispersion headWhat is certified in the displayed table? Take k=1 and h=1/4,1/8,1/16. For x=h/2∈[0,…
Exact mathematical passage
\lambda_{k,h}=\frac4{h^2}\sin^2\!\left(\frac{kh}{2}\right),\qquad h\ne0.
H^{-1}Use in source · 1 passage
Search aliases: H^-1
Use in source · Read passage · monograph_examples.tex, line 10
1 · 3) and retain its first coordinate. The eliminated block is invertible and the Schur complement of H is 2-1/3=5/3. Its inverse is 3/5, exactly the retained entry of H^-1. Inverting the compressed entry alone gives 1/2. The difference records the contribution of a path through the eliminated coordinate. Direct compression omits that contribution, while block inversion retains it. [H]centering includegraphics[width=.94textwidth]monograph_figures/exact_feshbach_compression.pdf captionThis ratio…
Exact mathematical passage
H^{-1}
h^{-2}Use in source · 1 passage
Search aliases: h^-2
Use in source · Read passage · 01_finite_core.tex, line 1143
[Low-mode second-order bound] For every real k and every nonzero real h, define λ_k,h=4h^-2sin^2(kh/2). Then 0≤ k^2-λ_k,h ≤ frack^4h^212.
Exact mathematical passage
\lambda_{k,h}=4h^{-2}\sin^2(kh/2)
h^2Use in source · 20 passages
Search aliases: h^2
Use in source · monograph_atlas.tex, line 461
vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters because the displayed expression divides by h^2. vfigdispersion headWhat is certified in the displayed table? Take k=1 and h=1/4,1/8,1/16. For x=h/2∈[0,…
Exact mathematical passage
\lambda_{k,h}=\frac4{h^2}\sin^2\!\left(\frac{kh}{2}\right),\qquad h\ne0.
H_0Use in source · 3 passages
Search aliases: H_0
Use in source · Read passage · 03_supplement_04.tex, line 40
Checked consequences. Unitarity also gives the capped estimate Qexp(-itH)P ≤min{1, |t| [H,P] }. If H=H_0+V, both summands are self-adjoint, and [H_0,P]=0, then Qexp(-it(H_0+V))P ≤2|t| V . The latter uses [V,P] ≤2 V . Its constant is not claimed to be optimal. Exact commutation gives zero leakage. The work does not include unbounded generators or time-dependent Hamiltonians.
Exact mathematical passage
H=H_0+V
H_1Use in source · 1 passage
Search aliases: H_1
Use in source · monograph_atlas.tex, line 766
vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and…
Exact mathematical passage
H_n=\diag(0,1,1/2,\ldots,1/(n-1)),\qquad H_1=(0).
\mathcal H_AUse in source · 1 passage
Search aliases: H_A mathcal
Use in source · Read passage · 03_supplement_03.tex, line 81
Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.
Exact mathematical passage
\boxed{\mathcal O_A(\psi)=\mathcal H_A\otimes\mathcal H_B \quad\Longleftrightarrow\quad \Ran\rho_B=\mathcal H_B.}
\mathcal H_BUse in source · 1 passage
Search aliases: H_B mathcal
Use in source · Read passage · 03_supplement_03.tex, line 81
Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.
Exact mathematical passage
\boxed{\mathcal O_A(\psi)=\mathcal H_A\otimes\mathcal H_B \quad\Longleftrightarrow\quad \Ran\rho_B=\mathcal H_B.}
H_{\mathrm{eff}}Use in source · 7 passages
Search aliases: H_eff mathrm
Use in source · Read passage · 01_finite_core.tex, line 907
[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the appropriately sized scalar matrices minus the indicated blocks.
Exact mathematical passage
\boxed{ H_{\mathrm{eff}}(z) = H_{PP} +H_{PQ}(z-H_{QQ})^{-1}H_{QP}. }
H_{\mathrm{eff}}^*Use in source · 1 passage
Search aliases: H_eff^* mathrm
Use in source · Read passage · 03_supplement_02.tex, line 110
The explicitly defined coordinate curve y(s)=U(s)^*ψ(s) is differentiable at t, with dot y=dot U^*ψ(t)+U(t)^*dotψ. It satisfies idot y=H_effy(t), H_eff=U(t)^*HU(t)-iU(t)^*dot U. The connection term -iU(t)^*dot U is Hermitian. If H^*=H, then H_eff^*=H_eff.
Exact mathematical passage
H_{\mathrm{eff}}^*=H_{\mathrm{eff}}
H_{\mathrm{even}}Use in source · 2 passages
Search aliases: H_even mathrm
Use in source · Read passage · 01_finite_core.tex, line 395
For two qubits and real parameters J,h, consider H = JX⊗ X +h(Z⊗ I+I⊗ Z). In the ordered even basis {ket00,ket11} and odd basis {ket01,ket10}, H_even = 2h · J J · -2h , H_odd = 0 · J J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.
Exact mathematical passage
H_{\mathrm{even}} = \begin{pmatrix} 2h&J\\ J&-2h \end{pmatrix}, \qquad H_{\mathrm{odd}} = \begin{pmatrix} 0&J\\ J&0 \end{pmatrix}.
H_nUse in source · 24 passages
Search aliases: H_n
Use in source · monograph_atlas.tex, line 766
vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and…
Exact mathematical passage
H_n=\diag(0,1,1/2,\ldots,1/(n-1)),\qquad H_1=(0).
H_{n+1}Use in source · 3 passages
Search aliases: H_n+1
Use in source · monograph_atlas.tex, line 769
vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and…
Exact mathematical passage
H_{n+1}R_n=R_nH_n
H_{n+1,n}Use in source · 1 passage
Search aliases: H_n+1,n
Use in source · Read passage · 01_finite_core.tex, line 1079
The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
Exact mathematical passage
H_{nn}=2\kappa+V(n), \qquad H_{n,n+1}=H_{n+1,n}=-\kappa,
H_{n,n+1}Use in source · 1 passage
Search aliases: H_n,n+1
Use in source · Read passage · 01_finite_core.tex, line 1079
The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
Exact mathematical passage
H_{nn}=2\kappa+V(n), \qquad H_{n,n+1}=H_{n+1,n}=-\kappa,
H_{nn}Use in source · 1 passage
Search aliases: H_nn
Use in source · Read passage · 01_finite_core.tex, line 1079
The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
Exact mathematical passage
H_{nn}=2\kappa+V(n), \qquad H_{n,n+1}=H_{n+1,n}=-\kappa,
H_{\mathrm{odd}}Use in source · 2 passages
Search aliases: H_odd mathrm
Use in source · Read passage · 01_finite_core.tex, line 395
For two qubits and real parameters J,h, consider H = JX⊗ X +h(Z⊗ I+I⊗ Z). In the ordered even basis {ket00,ket11} and odd basis {ket01,ket10}, H_even = 2h · J J · -2h , H_odd = 0 · J J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.
Exact mathematical passage
H_{\mathrm{even}} = \begin{pmatrix} 2h&J\\ J&-2h \end{pmatrix}, \qquad H_{\mathrm{odd}} = \begin{pmatrix} 0&J\\ J&0 \end{pmatrix}.
H_{PP}Use in source · 6 passages
Search aliases: H_PP
Use in source · monograph_atlas.tex, line 389
1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. This is the corrected hypothesis set recorded in the formal release. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, an error of 1/6. The omitted coupling has changed the response even though the retained diagonal entry did not change. headA pole can bel…
Exact mathematical passage
H_{PP}=0
H_{PQ}Use in source · 7 passages
Search aliases: H_PQ
Use in source · Read passage · 01_finite_core.tex, line 897
Let P be a retained packet projector and Q=I-P. In packet coordinates, H= H_PP · H_PQ H_QP · H_QQ .
Exact mathematical passage
H= \begin{pmatrix} H_{PP}&H_{PQ}\\ H_{QP}&H_{QQ} \end{pmatrix}.
H_{QP}Use in source · 7 passages
Search aliases: H_QP
Use in source · Read passage · 01_finite_core.tex, line 897
Let P be a retained packet projector and Q=I-P. In packet coordinates, H= H_PP · H_PQ H_QP · H_QQ .
Exact mathematical passage
H= \begin{pmatrix} H_{PP}&H_{PQ}\\ H_{QP}&H_{QQ} \end{pmatrix}.
H_{QQ}Use in source · 12 passages
Search aliases: H_QQ
Use in source · monograph_atlas.tex, line 381
1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. This is the corrected hypothesis set recorded in the formal release. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, an error of 1/6. The omitted coupling has changed the response even though the retained diagonal entry did not change. headA pole can bel…
Exact mathematical passage
H_{QQ}=2
H_{\rm d}Use in source · 4 passages
Search aliases: H_rm d rm
Use in source · Read passage · 02_strengthened.tex, line 576
Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.
Exact mathematical passage
H_{\rm d}=PHP+QHQ, \qquad H_{\rm o}=PHQ+QHP.
H_{\rm eff}Use in source · 1 passage
Search aliases: H_rm eff rm
Use in source · monograph_atlas.tex, line 382
1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. This is the corrected hypothesis set recorded in the formal release. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, an error of 1/6. The omitted coupling has changed the response even though the retained diagonal entry did not change. headA pole can bel…
Exact mathematical passage
H_{\rm eff}(z)=\frac1{z-2},\qquad \bigl[(zI-H)^{-1}\bigr]_{00} =\frac1{z-1/(z-2)}=\frac{z-2}{z^2-2z-1}.
H_{\rm even}Use in source · 1 passage
Search aliases: H_rm even rm
Use in source · monograph_atlas.tex, line 175
etry reveals blocks, not approximationsONE HAMILTONIAN IN TWO ORDERSTraditional edition: Theorem II.5.1. Take the exact rational Hamiltonian H=X⊗ X+frac38(Z⊗ I+I⊗ Z). It commutes with total parity Z⊗ Z. Reordering the computational basis into even addresses (00,11) and odd addresses (01,10) displays the invariant subspaces explicitly. vfighamiltonian H_rm even=mat3/4 · 1 1 · -3/4, H_rm odd=mat0 · 1 1 · 0. The even block squares to (25/16)I, so its eigenvalues are ±5/4. The odd block squares to I…
Exact mathematical passage
H_{\rm even}=\mat{3/4&1\\1&-3/4},\qquad H_{\rm odd}=\mat{0&1\\1&0}.
H_{\rm frame}Use in source · 1 passage
Search aliases: H_rm frame rm
Use in source · monograph_atlas.tex, line 218
1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i - i · 0, idot y=H_rm framey. This matrix is Hermitian. Omitting it would predict a constant y, contradicting the displayed coordinates. In this example the norm of the omitted time derivative is exactly one, so the failure is not a small numerical residual.
Exact mathematical passage
H_{\rm frame}=-\ii J=\mat{0&\ii\\-\ii&0}, \qquad \ii\dot y=H_{\rm frame}y.
H_{\rm Had}Use in source · 4 passages
Search aliases: H_rm Had rm
Use in source · monograph_atlas.tex, line 40
A Hamiltonian and a Hadamard gate are both traditionally denoted H. Where both appear, the gate is H_rm Had and the Hamiltonian keeps the letter H. Numerical labels on a drawing are coordinates or exact example values, not observations of a physical device. takeawayCheck the type before interpreting the picture. An idempotent projector, a density matrix, and a channel matrix have different defining equations.
Exact mathematical passage
H_{\rm Had}
H_{\rm mov}Use in source · 5 passages
Search aliases: H_rm mov rm
Use in source · Read passage · 02_strengthened.tex, line 704
[Moving frame Hamiltonian] Let U(t) solve idot U(t)=H(t)U(t), U(0)=I, where H(t)=H(t)^* is continuous on the compact time interval under consideration. Set V(t)=W(t)^*U(t). Then idot V(t)=H_rm mov(t)V(t), where H_rm mov(t) = W(t)^*H(t)W(t)- i W(t)^*dot W(t) = W(t)^*(H(t)- i K(t))W(t). The moving frame Hamiltonian is self adjoint.
Exact mathematical passage
\ii\dot V(t)=H_{\rm mov}(t)V(t),
H_{\rm o}Use in source · 3 passages
Search aliases: H_rm o rm
Use in source · Read passage · 02_strengthened.tex, line 576
Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.
Exact mathematical passage
H_{\rm d}=PHP+QHQ, \qquad H_{\rm o}=PHQ+QHP.
H_{\rm odd}Use in source · 1 passage
Search aliases: H_rm odd rm
Use in source · monograph_atlas.tex, line 175
etry reveals blocks, not approximationsONE HAMILTONIAN IN TWO ORDERSTraditional edition: Theorem II.5.1. Take the exact rational Hamiltonian H=X⊗ X+frac38(Z⊗ I+I⊗ Z). It commutes with total parity Z⊗ Z. Reordering the computational basis into even addresses (00,11) and odd addresses (01,10) displays the invariant subspaces explicitly. vfighamiltonian H_rm even=mat3/4 · 1 1 · -3/4, H_rm odd=mat0 · 1 1 · 0. The even block squares to (25/16)I, so its eigenvalues are ±5/4. The odd block squares to I…
Exact mathematical passage
H_{\rm even}=\mat{3/4&1\\1&-3/4},\qquad H_{\rm odd}=\mat{0&1\\1&0}.
H_TUse in source · 3 passages
Search aliases: H_T
Use in source · Read passage · 02_strengthened.tex, line 1860
[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.
Exact mathematical passage
H_T=-a^{-1}\log T
H_{\ell+1}Use in source · 1 passage
Search aliases: H_ℓ+1 ell
Use in source · Read passage · 02_strengthened.tex, line 2368
[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
H_{\ell+1}R_\ell=R_\ell H_\ell
H_\inftyUse in source · 7 passages
Search aliases: H_∞ infty
Use in source · monograph_atlas.tex, line 769
ditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and give a fixed complete system of two packets in the limit. xfigtower_tail headCompatibi…
Exact mathematical passage
H_\infty
\Hcal_AUse in source · 3 passages
Search aliases: Hcal_A H_A Hcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 154
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, let d_A=dim H_A≥1, and let r be the Schmidt rank of ψ. Then dimspan{(A⊗ I_B)ψ:A∈End( H_A)} = d_A r. Local operators on A generate the whole bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\psi\in\Hcal_A\otimes\Hcal_B
\Hcal_BUse in source · 1 passage
Search aliases: Hcal_B H_B Hcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 154
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, let d_A=dim H_A≥1, and let r be the Schmidt rank of ψ. Then dimspan{(A⊗ I_B)ψ:A∈End( H_A)} = d_A r. Local operators on A generate the whole bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\psi\in\Hcal_A\otimes\Hcal_B
\Hcal_{\phys}Use in source · 4 passages
Search aliases: Hcal_phys H_phys Hcal phys mathcal mathrm
Use in source · Read passage · 02_strengthened.tex, line 2216
[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.
Exact mathematical passage
\Hcal_{\phys}=\bigcap_a\Ker C_a
\Hcal_XUse in source · 2 passages
Search aliases: Hcal_X H_X Hcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 1042
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
\Phi:\End(\Hcal_X)\to\End(\Hcal_Y)
\Hcal_YUse in source · 3 passages
Search aliases: Hcal_Y H_Y Hcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 1042
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
\Phi:\End(\Hcal_X)\to\End(\Hcal_Y)
\Hcal_ZUse in source · 1 passage
Search aliases: Hcal_Z H_Z Hcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 1054
[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.
Exact mathematical passage
\Psi:\End(\Hcal_Y)\to\End(\Hcal_Z)
\Hcal_\inftyUse in source · 4 passages
Search aliases: Hcal_∞ H_∞ Hcal infty mathcal infty
Use in source · Read passage · 02_strengthened.tex, line 2360
Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.
Exact mathematical passage
\Hcal_\infty
\HHUse in source · 7 passages
Search aliases: HH
Use in source · monograph_atlas.tex, line 25
midrule HH,ψ · Hilbert space and state vector · They specify the space of addresses and its coherent amplitudes.
Exact mathematical passage
\HH,\psi
\HH_nUse in source · 1 passage
Search aliases: HH_n HH
Use in source · monograph_atlas.tex, line 765
vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and…
Exact mathematical passage
\HH_n=\CC^n
\hlineUse in source · 2 passages
Search aliases: hline
Use in source · monograph_atlas.tex, line 534
, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and three-cycle classes are c|rrr|r · e · (12) · (123) · d_λ trivial · 1 · 1 · 1 · 1 sign · 1 · -1 · 1 · 1 standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3…
Exact mathematical passage
\begin{array}{c|rrr|r} & e & (12)&(123)&d_\lambda\\\hline \text{trivial}&1&1&1&1\\ \text{sign}&1&-1&1&1\\ \text{standard}&2&0&-1&2 \end{array} \qquad P_\lambda=\frac{d_\lambda}{6}\sum_{g\in S_3}\chi_\lambda(g^{-1})U(g).
\HSUse in source · 4 passages
Search aliases: HS mathrm
Use in source · Read passage · 02_strengthened.tex, line 939
Density matrices and observables carry both an input packet and an output packet. Equip End( H) with the Hilbert Schmidt inner product ipXY_HS=Tr(X^*Y).
Exact mathematical passage
\ip{X}{Y}_{\HS}=\Tr(X^*Y).
I^*Use in source · 1 passage
Search aliases: I^*
Use in source · monograph_atlas.tex, line 370
main inside the original code, so the previous syndrome partition reports no bit flip. frac3ket000+4ket1115 ⟼ frac3ket000-4ket1115. The populations are unchanged, but the relative phase is not. vfiglogical_error Let V be the code isometry with columns ket000,ket111. For the proposed error family {I,Z_1}, its off-diagonal error compression is V^*I^*Z_1V=mat1 · 0 0 · -1. This matrix is not cI for any scalar c. Thus the Knill--Laflamme condition fails for that family. Its trace is zero even though …
Exact mathematical passage
V^*I^*Z_1V=\mat{1&0\\0&-1}.
I_3Use in source · 3 passages
Search aliases: I_3
Use in source · monograph_atlas.tex, line 74
vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0
Exact mathematical passage
M_2(\CC)\otimes I_3
I_6Use in source · 1 passage
Search aliases: I_6
Use in source · monograph_atlas.tex, line 544
standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3χ_λ(g^-1)U(g). All matrices in this example are rational. Multiplying them gives self-adjoint idempotents of ranks 1,1,4, with pairwise zero products and sum I_6. xfigs3 headThe standard packet has dimension four The standard representation occurs twice in the regular representation. Its isotypic component therefore has dimension 2·2=4: one factor carries the two-dimensional irreducible action and the other records multiplicity. A character projector…
Exact mathematical passage
I_6
I_BUse in source · 4 passages
Search aliases: I_B
Use in source · Read passage · 01_finite_core.tex, line 138
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\boxed{ \dim\operatorname{span}\{(A\otimes I_B)\psi:A\in M_{d_A}(\bbC)\} =d_A r. }
I_{\Hcal}Use in source · 1 passage
Search aliases: I_Hcal I_ H Hcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 215
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda, \qquad \sum_{\lambda\in\widehat G}P_\lambda=I_{\Hcal}.
I_JUse in source · 2 passages
Search aliases: I_J
Use in source · Read passage · 03_supplement_02.tex, line 211
To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive …
Exact mathematical passage
\boxed{\quad \Phi^{(J)}(X)=\sum_a (I_J\otimes K_a)X(I_J\otimes K_a)^*. \quad}
I_mUse in source · 7 passages
Search aliases: I_m
Use in source · monograph_atlas.tex, line 82
0 · 0 · 0. These two matrices provide a complete set of row directions. Their individual orbit dimensions are four and two, and together those spans fill the six-dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst bloc…
Exact mathematical passage
M_d(\CC)\otimes I_m
I_{m_r}Use in source · 1 passage
Search aliases: I_m_r
Use in source · Read passage · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
I_{M_\mu}Use in source · 1 passage
Search aliases: I_M_μ mu
Use in source · Read passage · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
I_PUse in source · 3 passages
Search aliases: I_P
Use in source · Read passage · 03_supplement_01.tex, line 101
Corrected Feshbach resolvent. Write a finite complex matrix in retained and eliminated coordinates: H=A · B{ C · D, zI-H=zI_P-A · -B -C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).
Exact mathematical passage
H=\begin{pmatrix}A&B\\C&D\end{pmatrix},\qquad zI-H=\begin{pmatrix}zI_P-A&-B\\-C&zI_Q-D\end{pmatrix}.
I_QUse in source · 3 passages
Search aliases: I_Q
Use in source · Read passage · 01_finite_core.tex, line 906
[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the appropriately sized scalar matrices minus the indicated blocks.
Exact mathematical passage
zI_Q-H_{QQ}
I_{\rm in}Use in source · 1 passage
Search aliases: I_rm in rm
Use in source · Read passage · 02_strengthened.tex, line 1083
labelstrong:standardfact:001 A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.
Exact mathematical passage
\Tr_{\rm out}J(\Phi)=I_{\rm in}.
\idUse in source · 6 passages
Search aliases: id mathrm
Use in source · monograph_atlas.tex, line 330
vpage18An ancilla tests more than positivityWHY COMPLETE POSITIVITY IS THE RELEVANT CONDITIONTraditional edition: Classical input II.7.7. Ordinary transpose T sends a positive matrix on one system to another positive matrix and preserves its trace. Those properties are insufficient for a quantum channel on a subsystem of an entangled state. The test must include id⊗ T on the larger system. vfigcomplete_positivity Apply this extended map to the Bell density matrix. The result is (id⊗ T)(ρ_rm Bell…
Exact mathematical passage
\id\otimes\mathsf T
\ii^{-jk}Use in source · 1 passage
Search aliases: ii^-jk i^-jk ii mathrm
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
\ii^{-jr}Use in source · 1 passage
Search aliases: ii^-jr i^-jr ii mathrm
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
\ii^jUse in source · 1 passage
Search aliases: ii^j i^j ii mathrm
Use in source · monograph_atlas.tex, line 100
character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing the address multiplies it by i^j. The entries show phase differences rather than four different probability distributions: every column has uniform site probability 1/4.
Exact mathematical passage
\ii^j
\ImUse in source · 1 passage
Search aliases: Im
Use in source · monograph_atlas.tex, line 653
1 · 2, so A=0, B=1, and D=2. In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. …
Exact mathematical passage
\Sigma(3+\ii)=\frac{1-\ii}{2},\qquad \Im\Sigma(z)=-(\Im z)\left|(z-2)^{-1}\right|^2\leq0.
\operatorname{Im}Use in source · 2 passages
Search aliases: Im operatorname
Use in source · Read passage · 02_strengthened.tex, line 1576
[Upper half plane sign] If Imz>0, then ImΣ(z) = -(Imz) B(z-D)^-1(overline z-D)^-1B^* ≤0. Thus -Σ is an operator valued Herglotz function.
Exact mathematical passage
\operatorname{Im}z>0
\operatorname{im}Use in source · 1 passage
Search aliases: im operatorname
Use in source · constraint_image.tex, line 8
same constraints;node[rotate=90,font=footnotesize] at (11.4,1.7) new coordinate; node[font=small] at (6,0.15) closure on imR transfers, and the new complement needs its own test;
Exact mathematical passage
\operatorname{im}R
\ipUse in source · 11 passages
Search aliases: ip
Use in source · Read passage · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
JUse in source · 20 passages
Search aliases: J
Use in source · monograph_atlas.tex, line 217
-sin t. The vector has not moved in the laboratory, but its coordinate entries have changed. vfigmoving_frame headThe missing term can be computed exactly Let J=mat0 · -1 1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i
Exact mathematical passage
J=\mat{0&-1\\1&0}
\bm jUse in source · 1 passage
Search aliases: j bm
Use in source · Read passage · 02_strengthened.tex, line 343
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
P_{\bm j}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
\mathbf jUse in source · 1 passage
Search aliases: j mathbf
Use in source · Read passage · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
J^2Use in source · 1 passage
Search aliases: J^2
Use in source · Read passage · 01_finite_core.tex, line 411
J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.
Exact mathematical passage
\Spec H_{\mathrm{even}} =\{\pm\sqrt{4h^2+J^2}\}, \qquad \Spec H_{\mathrm{odd}}=\{\pm J\}.
j_1Use in source · 4 passages
Search aliases: j_1
Use in source · Read passage · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
j_mUse in source · 4 passages
Search aliases: j_m
Use in source · Read passage · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
j_rUse in source · 1 passage
Search aliases: j_r
Use in source · Read passage · 02_strengthened.tex, line 2476
[Reflection positivity is closed under limits] Suppose C_n(j,k) are reflection positive kernels on N, meaning every finite matrix (C_n(j_r,j_s))_r,s is positive semidefinite. If C_n(j,k)⟶ C(j,k) for every pair, then C is reflection positive.
Exact mathematical passage
\bigl(C_n(j_r,j_s)\bigr)_{r,s}
j_sUse in source · 1 passage
Search aliases: j_s
Use in source · Read passage · 02_strengthened.tex, line 2476
[Reflection positivity is closed under limits] Suppose C_n(j,k) are reflection positive kernels on N, meaning every finite matrix (C_n(j_r,j_s))_r,s is positive semidefinite. If C_n(j,k)⟶ C(j,k) for every pair, then C is reflection positive.
Exact mathematical passage
\bigl(C_n(j_r,j_s)\bigr)_{r,s}
K'Use in source · 1 passage
Search aliases: K'
Use in source · monograph_atlas.tex, line 442
vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESTraditional edition: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect…
Exact mathematical passage
K(\tau)=\frac{1+2e^{-2\tau}+e^{-4\tau}}4 =\frac{(1+e^{-2\tau})^2}4, \qquad d_s(\tau)=-2\tau\frac{K'(\tau)}{K(\tau)} =\frac{8\tau}{e^{2\tau}+1}.
K^*Use in source · 1 passage
Search aliases: K^*
Use in source · Read passage · 02_strengthened.tex, line 719
Differentiate V=W^*U. Since dot W=KW and K^*=-K, dot W^*=-W^*K. Substitution gives the equation. The first term in H_rm mov is self adjoint. The operator - i W^*dot W is self adjoint because W^*dot W is skew adjoint.
Exact mathematical passage
K^*=-K
K^2Use in source · 1 passage
Search aliases: K^2
Use in source · monograph_atlas.tex, line 454
For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.
Exact mathematical passage
K_2=K^2
k^2Use in source · 9 passages
Search aliases: k^2
Use in source · monograph_atlas.tex, line 460
vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters because the displayed expression divides by h^2. vfigdispersion headWhat is certified in the displayed table? Take k=1 and h=1/4,1/8,1/16. For x=h/2∈[0,…
Exact mathematical passage
k^2
k^4Use in source · 6 passages
Search aliases: k^4
Use in source · monograph_atlas.tex, line 464
vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters because the displayed expression divides by h^2. vfigdispersion headWhat is certified in the displayed table? Take k=1 and h=1/4,1/8,1/16. For x=h/2∈[0,…
Exact mathematical passage
0\le k^2-\lambda_{k,h}\le k^4h^2/12
K_0Use in source · 5 passages
Search aliases: K_0
Use in source · monograph_atlas.tex, line 312
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state…
Exact mathematical passage
K_0=\mat{1&0\\0&4/5},\qquad K_1=\mat{0&3/5\\0&0},\qquad K_0^*K_0+K_1^*K_1=I.
K_0^*Use in source · 2 passages
Search aliases: K_0^*
Use in source · monograph_atlas.tex, line 312
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state…
Exact mathematical passage
K_0=\mat{1&0\\0&4/5},\qquad K_1=\mat{0&3/5\\0&0},\qquad K_0^*K_0+K_1^*K_1=I.
K_1Use in source · 6 passages
Search aliases: K_1
Use in source · monograph_atlas.tex, line 312
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state…
Exact mathematical passage
K_0=\mat{1&0\\0&4/5},\qquad K_1=\mat{0&3/5\\0&0},\qquad K_0^*K_0+K_1^*K_1=I.
K_1^*Use in source · 2 passages
Search aliases: K_1^*
Use in source · monograph_atlas.tex, line 312
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state…
Exact mathematical passage
K_0=\mat{1&0\\0&4/5},\qquad K_1=\mat{0&3/5\\0&0},\qquad K_0^*K_0+K_1^*K_1=I.
K_2Use in source · 1 passage
Search aliases: K_2
Use in source · monograph_atlas.tex, line 454
For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.
Exact mathematical passage
K_2=K^2
K_aUse in source · 13 passages
Search aliases: K_a
Use in source · monograph_atlas.tex, line 30
K_a · Kraus operator · It describes one channel branch and need not be a projector.
Exact mathematical passage
K_a
K_a^*Use in source · 8 passages
Search aliases: K_a^*
Use in source · monograph_atlas.tex, line 316
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
\Phi(\rho)=\sum_aK_a\rho K_a^*
K_jUse in source · 6 passages
Search aliases: K_j
Use in source · Read passage · monograph_examples.tex, line 17
e, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to th…
Exact mathematical passage
\sum_j K_j^*K_j=I
k_jUse in source · 2 passages
Search aliases: k_j
Use in source · Read passage · 02_strengthened.tex, line 2022
Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].
Exact mathematical passage
\lambda_N(k) = 4N^2 \sum_{j=1}^{d} \sin^2\left(\frac{\pi k_j}{N}\right),
K_j^*Use in source · 2 passages
Search aliases: K_j^*
Use in source · Read passage · monograph_examples.tex, line 17
e, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to th…
Exact mathematical passage
\sum_j K_j^*K_j=I
k_j^4Use in source · 1 passage
Search aliases: k_j^4
Use in source · Read passage · 02_strengthened.tex, line 2033
[Low mode dispersion in dimension d] For every centered mode k, 0 ≤ 4π^2absk^2-λ_N(k) ≤ frac4π^43N^2 ∑_j=1^dk_j^4.
Exact mathematical passage
0 \leq 4\pi^2\abs{k}^2-\lambda_N(k) \leq \frac{4\pi^4}{3N^2} \sum_{j=1}^{d}k_j^4.
K_rUse in source · 5 passages
Search aliases: K_r
Use in source · Read passage · 01_finite_core.tex, line 723
[Finite Kraus channel] Let K_1,…,K_r satisfy ∑_aK_a^*K_a=I. Then Φ(ρ)=∑_aK_aρ K_a^* is completely positive and trace preserving.
Exact mathematical passage
K_1,\ldots,K_r
K_r^*Use in source · 3 passages
Search aliases: K_r^*
Use in source · Read passage · 01_finite_core.tex, line 855
_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP from the F_r. Every recovery branch of a coherent combination is therefore scala…
Exact mathematical passage
\sum_rK_r^*K_r+K_{\mathrm{unused}}^*K_{\mathrm{unused}} =S+(I-S)^2=I.
K_{\mathrm{unused}}Use in source · 2 passages
Search aliases: K_unused mathrm
Use in source · Read passage · 01_finite_core.tex, line 851
efine compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP…
Exact mathematical passage
K_r=V_r^*,\qquad K_{\mathrm{unused}}=I-S.
K_{\mathrm{unused}}^*Use in source · 1 passage
Search aliases: K_unused^* mathrm
Use in source · Read passage · 01_finite_core.tex, line 855
_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP from the F_r. Every recovery branch of a coherent combination is therefore scala…
Exact mathematical passage
\sum_rK_r^*K_r+K_{\mathrm{unused}}^*K_{\mathrm{unused}} =S+(I-S)^2=I.
\KerUse in source · 4 passages
Search aliases: Ker operatorname
Use in source · Read passage · 02_strengthened.tex, line 1555
(z-D)^-1B^*x is an isomorphism from Ker F_P(z) onto Ker(z-H).
Exact mathematical passage
\Ker F_P(z)
\mathcal LUse in source · 1 passage
Search aliases: L mathcal
Use in source · Read passage · 03_supplement_04.tex, line 49
Actual projected differentiation and curvature. Let E,F be real normed spaces. Let P:E→ L(F,F) be differentiable with P(x)^2=P(x) for every x. Let s:E→ F be C^2 with P(x)s(x)=s(x), and let X,Y:E→ E be differentiable vector fields. The checked connection is the actual expression ∇_Xs(x)=P(x) Ds(x)[X(x)]. It is proved to preserve the range condition. It is linear in the vector field direction and obeys the scalar Leibniz rule ∇_X(cs)=Dc[X] s+c∇_Xs for differentiable real scalar functions c. No dif…
Exact mathematical passage
P:E\to\mathcal L(F,F)
L_jUse in source · 3 passages
Search aliases: L_j
Use in source · Read passage · 01_finite_core.tex, line 823
textitNecessity. Fix an error E_a and let L_j=K_jE_aP. Perfect recovery on rank-one code operators gives ∑_jketL_jψbraL_jψ =t_aketψbraψ (ψ∈ C). For every vector orthogonal to ψ, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous fo…
Exact mathematical passage
L_j=K_jE_aP
L_NUse in source · 1 passage
Search aliases: L_N
Use in source · Read passage · 02_strengthened.tex, line 2153
[Rigorous intermediate dimension window] For every ε>0, there is a diffusion time t_ε>0 and an integer N_ε such that absd_L_N(t_ε)-d<ε for every N≥ N_ε.
Exact mathematical passage
\abs{d_{L_N}(t_\varepsilon)-d}<\varepsilon
L_nUse in source · 4 passages
Search aliases: L_n
Use in source · Read passage · 02_strengthened.tex, line 2511
[Heat trace convergence from trace norm] Let L_n,L be nonnegative self adjoint operators on one Hilbert space. For a fixed t>0, assume their heat operators are trace class and that e^-tL_n⟶ e^-tL in trace norm. Then Tr( e^-tL_n)⟶Tr( e^-tL). If also the heat moment operators are trace class and L_n e^-tL_n⟶ L e^-tL in trace norm and the limiting heat trace is nonzero, then the spectral dimensions defined by the heat moment ratio converge at that time.
Exact mathematical passage
L_n,L
L_rUse in source · 4 passages
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Use in source · Read passage · 02_strengthened.tex, line 1214
[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace preserving semigroup [source reference].
Exact mathematical passage
\Gcal(X) = -\ii[H,X] + \sum_r \left( L_rXL_r^* - \frac12L_r^*L_rX - \frac12XL_r^*L_r \right),
L_r^*Use in source · 3 passages
Search aliases: L_r^*
Use in source · Read passage · 02_strengthened.tex, line 1214
[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace preserving semigroup [source reference].
Exact mathematical passage
\Gcal(X) = -\ii[H,X] + \sum_r \left( L_rXL_r^* - \frac12L_r^*L_rX - \frac12XL_r^*L_r \right),
\LcalUse in source · 9 passages
Search aliases: Lcal L mathcal
Use in source · monograph_atlas.tex, line 580
vpage31Coherence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSTraditional edition: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space are L_0(X)=(X+ZXZ)/2 and L_1(X)=(X-ZXZ)/2.
Exact mathematical passage
\Lcal_0(X)=(X+ZXZ)/2
\Lcal_0Use in source · 1 passage
Search aliases: Lcal_0 L_0 Lcal mathcal
Use in source · monograph_atlas.tex, line 580
vpage31Coherence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSTraditional edition: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space are L_0(X)=(X+ZXZ)/2 and L_1(X)=(X-ZXZ)/2.
Exact mathematical passage
\Lcal_0(X)=(X+ZXZ)/2
\Lcal_1Use in source · 1 passage
Search aliases: Lcal_1 L_1 Lcal mathcal
Use in source · monograph_atlas.tex, line 580
er survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSTraditional edition: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space are L_0(X)=(X+ZXZ)/2 and L_1(X)=(X-ZXZ)/2.
Exact mathematical passage
\Lcal_1(X)=(X-ZXZ)/2
\Lcal_pUse in source · 2 passages
Search aliases: Lcal_p L_p Lcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 1241
[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.
Exact mathematical passage
\Gcal\Lcal_p=\Lcal_p\Gcal.
\Lcal_qUse in source · 5 passages
Search aliases: Lcal_q L_q Lcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 1099
[Coherence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.
Exact mathematical passage
\Lcal_q(X) = \frac1N \sum_{r=0}^{N-1} \omega^{-qr}S^rXS^{-r}.
\limUse in source · 2 passages
Search aliases: lim
Use in source · Read passage · 02_strengthened.tex, line 1972
[Finite scale limits] For every finite nonnegative L with nontrivial kernel, lim_tdownarrow0d_L(t)=0 and lim_t→∞d_L(t)=0. For every t>0, 0≤ d_L(t)≤2tnorm L.
Exact mathematical passage
\lim_{t\downarrow0}d_L(t)=0
\lim_{t\downarrow0}Use in source · 1 passage
Search aliases: lim_tdownarrow0 lim downarrow
Use in source · Read passage · 02_strengthened.tex, line 1972
[Finite scale limits] For every finite nonnegative L with nontrivial kernel, lim_tdownarrow0d_L(t)=0 and lim_t→∞d_L(t)=0. For every t>0, 0≤ d_L(t)≤2tnorm L.
Exact mathematical passage
\lim_{t\downarrow0}d_L(t)=0
\lim_{t\to\infty}Use in source · 1 passage
Search aliases: lim_t→∞ lim to infty
Use in source · Read passage · 02_strengthened.tex, line 1976
[Finite scale limits] For every finite nonnegative L with nontrivial kernel, lim_tdownarrow0d_L(t)=0 and lim_t→∞d_L(t)=0. For every t>0, 0≤ d_L(t)≤2tnorm L.
Exact mathematical passage
\lim_{t\to\infty}d_L(t)=0.
\log_{10}Use in source · 1 passage
Search aliases: log_10 log
Use in source · spectral_dimension.tex, line 26
125); node[font=small,align=center,anchor=east,font=scriptsize] at (0.64,2.1125) 1; draw[black!13,line width=.25pt] (0.8,3.425) -- (9.5,3.425); node[font=small,align=center,anchor=east,font=scriptsize] at (0.64,3.425) 2; draw[->,line width=.6pt] (0.8,0.8) -- (9.62,0.8); draw[->,line width=.6pt] (0.8,0.8) -- (0.8,4.07); node[font=small,align=center,font=scriptsize] at (5.15,0.25) log_10τ; node[font=small,align=center,anchor=west,font=scriptsize] at (0.8,4.35) d_s(τ); draw[line width=1.1pt] plot c…
Exact mathematical passage
\log_{10}\tau
\LongleftrightarrowUse in source · 6 passages
Search aliases: Longleftrightarrow
Use in source · Read passage · 02_strengthened.tex, line 1542
[Spectral equivalence] Let z∉Spec D. Then z∈Spec H Longleftrightarrow 0∈Spec F_P(z). The map x⟼ x
Exact mathematical passage
z\in\Spec H \quad\Longleftrightarrow\quad 0\in\Spec F_P(z).
\LongrightarrowUse in source · 3 passages
Search aliases: Longrightarrow
Use in source · Read passage · 01_finite_core.tex, line 803
We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix A_w.
Exact mathematical passage
PXP=X\quad\Longrightarrow\quad \cR(A_wXA_w^*)=t_wX.
M_2Use in source · 2 passages
Search aliases: M_2
Use in source · monograph_atlas.tex, line 74
vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0
Exact mathematical passage
M_2(\CC)\otimes I_3
M_dUse in source · 8 passages
Search aliases: M_d
Use in source · monograph_atlas.tex, line 61
ment changes the local spanSCHMIDT RANK AS ACCESSIBLE MULTIPLICITYTraditional edition: Theorem II.2.2, Corollary II.2.4. Write a bipartite vector as a coefficient matrix C, with the first subsystem indexing rows. Acting only on that subsystem transforms C to AC. Every resulting row remains in the row space of C, and every independent choice of such rows can be obtained in this way. dimSpan{(A⊗ I)ψ:A∈ M_d( C)} =d rank C. vfiglocal_orbits The product vector ket00 has matrix C=diag(1,0). Its local …
Exact mathematical passage
\dim\Span\{(A\otimes I)\psi:A\in M_d(\CC)\} =d\,\rank C.
M_{d_A}Use in source · 2 passages
Search aliases: M_d_A
Use in source · Read passage · 01_finite_core.tex, line 138
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\boxed{ \dim\operatorname{span}\{(A\otimes I_B)\psi:A\in M_{d_A}(\bbC)\} =d_A r. }
M_{d_r}Use in source · 1 passage
Search aliases: M_d_r
Use in source · Read passage · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
M_{d\times m}Use in source · 2 passages
Search aliases: M_d× m times
Use in source · Read passage · 03_supplement_03.tex, line 2
Actual orbits and sharp seed counts. Let C∈ M_d× m( C) be the coefficient matrix of one seed. Define its orbit by actual matrix multiplication: O(C)={AC:A∈ M_d( C)}. This set is already a linear subspace. The source proves that it equals the linear span of the orbit before computing its dimension.
Exact mathematical passage
C\in M_{d\times m}(\C)
M_mUse in source · 1 passage
Search aliases: M_m
Use in source · Read passage · 02_strengthened.tex, line 1068
[Finite Kraus and Choi criteria] A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.
Exact mathematical passage
\Phi:M_n(\CC)\to M_m(\CC)
M_nUse in source · 1 passage
Search aliases: M_n
Use in source · Read passage · 02_strengthened.tex, line 1068
[Finite Kraus and Choi criteria] A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.
Exact mathematical passage
\Phi:M_n(\CC)\to M_m(\CC)
m_rUse in source · 7 passages
Search aliases: m_r
Use in source · Read passage · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
M_\muUse in source · 1 passage
Search aliases: M_μ mu
Use in source · Read passage · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
\matUse in source · 21 passages
Search aliases: mat
Use in source · monograph_atlas.tex, line 75
vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0 0 · 1 · 0, C_2=mat0 · 0 · 1 0 · 0 · 0. These two matrices provide a complete set of row directions. Their individual orbit dimensions are four and two, and …
Exact mathematical passage
C_1=\mat{1&0&0\\0&1&0},\qquad C_2=\mat{0&0&1\\0&0&0}.
\mathbinUse in source · 2 passages
Search aliases: mathbin
Use in source · Read passage · 01_finite_core.tex, line 647
n convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calculation is not a separate entry among the …
Exact mathematical passage
[\partial_\theta P,\partial_\phi P] =\frac{i}{2}(\partial_\theta\mathbf n\mathbin{\times}\partial_\phi\mathbf n)\cdot\boldsymbol\sigma.
\max_{1\le r\le s}Use in source · 1 passage
Search aliases: max_1≤ r≤ s max le le
Use in source · Read passage · 02_strengthened.tex, line 135
[Quantum seed rank] For the displayed multiplicity decomposition, srank_ A( H) = max_1≤ r≤ s lceilfracm_rd_rrceil.
Exact mathematical passage
\operatorname{srank}_{\Acal}(\Hcal) = \max_{1\le r\le s} \left\lceil\frac{m_r}{d_r}\right\rceil.
\max_rUse in source · 1 passage
Search aliases: max_r max
Use in source · Read passage · 02_strengthened.tex, line 147
Set k=max_rlceilfracm_rd_rrceil. For each r, partition a basis of C^m_r into at most k groups of size at most d_r. Choose the coefficient matrix of the jth seed so that its row space spans the jth group. The combined row spaces span every multiplicity factor. The preceding theorem then shows that the k seeds generate every summand.
Exact mathematical passage
k=\max_r\left\lceil\frac{m_r}{d_r}\right\rceil.
\minUse in source · 7 passages
Search aliases: min
Use in source · monograph_atlas.tex, line 198
in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε t=π/6, the amplitude is 1/2 and the probability is 1/4. Near zero, amplitudes are first order while escape probabilities are second order. Mixing thos…
Exact mathematical passage
\Pr(\text{hidden packet at }t)\le\min\{1,\varepsilon^2t^2\}.
\bm nUse in source · 3 passages
Search aliases: n bm
Use in source · Read passage · 02_strengthened.tex, line 891
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
P(\theta,\phi) = \frac12\left(I+\bm n(\theta,\phi)\cdot\bm\sigma\right),
\mathbf nUse in source · 5 passages
Search aliases: n mathbf
Use in source · Read passage · 01_finite_core.tex, line 637
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
P(\theta,\phi) = \frac12\left(I+\mathbf n(\theta,\phi)\cdot\boldsymbol\sigma\right),
\mathcal NUse in source · 5 passages
Search aliases: N mathcal
Use in source · Read passage · 03_supplement_05.tex, line 3
The precise finite channel statement. Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^*R_r=I. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire …
Exact mathematical passage
\mathcal N_E(\rho)=\sum_a E_a\rho E_a^*.
N^2Use in source · 4 passages
Search aliases: N^2
Use in source · monograph_atlas.tex, line 725
vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSTraditional edition: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).
Exact mathematical passage
\lambda_N(k)=4N^2\sin^2(\pi k/N)
n_1Use in source · 1 passage
Search aliases: n_1
Use in source · Read passage · 02_strengthened.tex, line 1875
[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.
Exact mathematical passage
n_1,\ldots,n_m
\mathcal N_EUse in source · 5 passages
Search aliases: N_E mathcal
Use in source · Read passage · 03_supplement_05.tex, line 3
The precise finite channel statement. Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^*R_r=I. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire …
Exact mathematical passage
\mathcal N_E(\rho)=\sum_a E_a\rho E_a^*.
n_jUse in source · 6 passages
Search aliases: n_j
Use in source · monograph_atlas.tex, line 716
xponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This is a finite Gram factorization. The …
Exact mathematical passage
n_j
n_kUse in source · 6 passages
Search aliases: n_k
Use in source · monograph_atlas.tex, line 717
tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This is a finite Gram factorization. The retained run checks the integer sem…
Exact mathematical passage
\sum_{j,k}\langle u_j,T^{n_j+n_k}u_k\rangle =\left\|\sum_jT^{n_j}u_j\right\|^2.
n_mUse in source · 1 passage
Search aliases: n_m
Use in source · Read passage · 02_strengthened.tex, line 1875
[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.
Exact mathematical passage
n_1,\ldots,n_m
N_\varepsilonUse in source · 5 passages
Search aliases: N_ε varepsilon
Use in source · monograph_atlas.tex, line 738
· 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less than ε/2. Then choose N_ε so that every N≥ N_ε differs from that continuum value by less than ε/2 at the same time. This proves one controlled intermediate time for all sufficient…
Exact mathematical passage
N_\varepsilon
\NNUse in source · 3 passages
Search aliases: NN N mathbb
Use in source · monograph_atlas.tex, line 769
a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and give a fixed complet…
Exact mathematical passage
\ell^2(\NN_0)
\NN_0Use in source · 1 passage
Search aliases: NN_0 N_0 NN mathbb
Use in source · monograph_atlas.tex, line 769
a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and give a fixed complet…
Exact mathematical passage
\ell^2(\NN_0)
OUse in source · 10 passages
Search aliases: O
Use in source · monograph_atlas.tex, line 126
vpage8Track the Bell state through the circuitINTERMEDIATE PACKETS ARE NOT FINAL PACKETSTraditional edition: Theorem II.4.1, Example II.4.6. The input is ket00. First apply a Hadamard to the first qubit, then a controlled flip: ket00 xmapstoH_rm Had⊗ I fracket00+ket10sqrt2 xmapstorm CNOT fracket00+ket11sqrt2. The displayed columns keep amplitudes, including their normalization, at every step. The middle vector occupies both parity packets. The final vector occupies only the even packet. vfigbell…
Exact mathematical passage
\ket{00}\ \xmapsto{H_{\rm Had}\otimes I}\ \frac{\ket{00}+\ket{10}}{\sqrt2} \ \xmapsto{\rm CNOT}\ \frac{\ket{00}+\ket{11}}{\sqrt2}.
\mathcal OUse in source · 4 passages
Search aliases: O mathcal
Use in source · Read passage · 03_supplement_03.tex, line 3
Actual orbits and sharp seed counts. Let C∈ M_d× m( C) be the coefficient matrix of one seed. Define its orbit by actual matrix multiplication: O(C)={AC:A∈ M_d( C)}. This set is already a linear subspace. The source proves that it equals the linear span of the orbit before computing its dimension.
Exact mathematical passage
\mathcal O(C)=\{AC:A\in M_d(\C)\}.
oUse in source · 19 passages
Search aliases: o
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigpar…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
O_1Use in source · 1 passage
Search aliases: O_1
Use in source · Read passage · 01_finite_core.tex, line 58
The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.
Exact mathematical passage
\cH(\psi_0;\cO) = \operatorname{span} \left\{ O_r\cdots O_1\psi_0: O_j\in\cO, \ r\ge0 \right\}.
\mathcal O_AUse in source · 1 passage
Search aliases: O_A mathcal
Use in source · Read passage · 03_supplement_03.tex, line 81
Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.
Exact mathematical passage
\boxed{\mathcal O_A(\psi)=\mathcal H_A\otimes\mathcal H_B \quad\Longleftrightarrow\quad \Ran\rho_B=\mathcal H_B.}
O_jUse in source · 1 passage
Search aliases: O_j
Use in source · Read passage · 01_finite_core.tex, line 58
The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.
Exact mathematical passage
\cH(\psi_0;\cO) = \operatorname{span} \left\{ O_r\cdots O_1\psi_0: O_j\in\cO, \ r\ge0 \right\}.
O_rUse in source · 1 passage
Search aliases: O_r
Use in source · Read passage · 01_finite_core.tex, line 58
The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.
Exact mathematical passage
\cH(\psi_0;\cO) = \operatorname{span} \left\{ O_r\cdots O_1\psi_0: O_j\in\cO, \ r\ge0 \right\}.
\mathrm{odd}Use in source · 3 passages
Search aliases: odd mathrm
Use in source · Read passage · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
\overline pUse in source · 1 passage
Search aliases: overline p overline
Use in source · Read passage · 03_supplement_05.tex, line 98
Rank-one rigidity of the corrected channel. The module tests the map equality on actual matrix units and flattens each A_ra into a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzero p, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication X…
Exact mathematical passage
X^*X=\overline p\,p^{\mathsf T}.
\overline UUse in source · 4 passages
Search aliases: overline U overline
Use in source · Read passage · 02_strengthened.tex, line 37
form a complete orthogonal packet system on Hilbert Schmidt operator space. Dephasing is the diagonal projection. Every channel has input and output superblocks, with the same composition law as ordinary matrix blocks. A covariant channel preserves the isotypic packets of the conjugation action. In the cyclic case these are coherence orders. The Choi matrix carries the equivalent symmetry under U⊗overline U.
Exact mathematical passage
U\otimes\overline U
\overline vUse in source · 1 passage
Search aliases: overline v overline
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
\overline v_jUse in source · 1 passage
Search aliases: overline v_j overline
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
\overline zUse in source · 2 passages
Search aliases: overline z overline
Use in source · Read passage · 02_strengthened.tex, line 1577
[Upper half plane sign] If Imz>0, then ImΣ(z) = -(Imz) B(z-D)^-1(overline z-D)^-1B^* ≤0. Thus -Σ is an operator valued Herglotz function.
Exact mathematical passage
\operatorname{Im}\Sigma(z) = -(\operatorname{Im}z) B(z-D)^{-1}(\overline z-D)^{-1}B^* \leq0.
\overline{c_b}Use in source · 1 passage
Search aliases: overlinec_b overline
Use in source · Read passage · 03_supplement_05.tex, line 175
For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.
Exact mathematical passage
\mathcal R\bigl(E(c)\rho E(c)^*\bigr) =\left(\sum_{a,b}c_a\overline{c_b}C_{ba}\right)\rho =(c^*Cc)\rho.
\overline{c_j}Use in source · 1 passage
Search aliases: overlinec_j overline
Use in source · Read passage · 02_strengthened.tex, line 1880
[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.
Exact mathematical passage
\sum_{j,k}\overline{c_j}c_k \ip{v_j}{T^{n_j+n_k}v_k} \geq0
\overline{d_{ja}}Use in source · 1 passage
Search aliases: overlined_ja overline
Use in source · Read passage · 01_finite_core.tex, line 833
he left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recovery completeness now gives PE_a^*E_bP · =∑_j (K_jE_aP)^*(K_jE_bP) · =(∑_joverlined_jad_jb)P. The coefficients form a Gram matrix, hence are positive se…
Exact mathematical passage
\begin{aligned} PE_a^*E_bP &=\sum_j (K_jE_aP)^*(K_jE_bP)\\ &=\left(\sum_j\overline{d_{ja}}d_{jb}\right)P. \end{aligned}
\overline{U(g)}Use in source · 1 passage
Search aliases: overlineU(g) overline
Use in source · Read passage · 02_strengthened.tex, line 1146
[Choi symmetry criterion] Let U:G→ U( H) be a finite group representation. A channel Φ is covariant, Φ(U(g)XU(g)^*) = U(g)Φ(X)U(g)^*, if and only if (U(g)⊗overlineU(g)) J(Φ) (U(g)^*⊗ U(g)^ T) = J(Φ) for every g∈ G.
Exact mathematical passage
\bigl(U(g)\otimes\overline{U(g)}\bigr) J(\Phi) \bigl(U(g)^*\otimes U(g)^{\mathsf T}\bigr) = J(\Phi)
\overline{W_{ja}}Use in source · 1 passage
Search aliases: overlineW_ja overline
Use in source · Read passage · 02_strengthened.tex, line 1323
[Orthogonal syndrome construction] Assume the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
F_j=\sum_a\overline{W_{ja}}E_a.
\overline{z_a}Use in source · 1 passage
Search aliases: overlinez_a overline
Use in source · Read passage · 03_supplement_04.tex, line 131
identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent superpositions. This construction supplies the recovery stage used in the full Knill--Laflamme proof that foll…
Exact mathematical passage
\mathcal R(E_z\rho E_z^*)=\left(\sum_a z_a\overline{z_a}\right)\rho.
\overline{z_{ra}}Use in source · 1 passage
Search aliases: overlinez_ra overline
Use in source · Read passage · 03_supplement_05.tex, line 111
Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this calculation with the original error and recovery matrices.
Exact mathematical passage
PE_a^*E_bP &=PE_a^*\Bigl(\sum_rR_r^*R_r\Bigr)E_bP\\ &=\sum_r(R_rE_aP)^*(R_rE_bP)\\ &=\Bigl(\sum_r\overline{z_{ra}}z_{rb}\Bigr)P =(Z^*Z)_{ab}P.
\overline{\chi(r)}Use in source · 1 passage
Search aliases: overlineχ(r) overline chi
Use in source · Read passage · 03_supplement_03.tex, line 126
Orthogonality of the projectors. Unitarity supplies ρ(r)^*=ρ(-r), while the character obeys overlineχ(r)=χ(-r). Reindexing r↦-r proves P_j^*=P_j. The final concrete matrix theorem includes Hermiticity together with multiplication, completeness, and eigenaction. Together these identities establish orthogonal spectral packets.
Exact mathematical passage
\overline{\chi(r)}=\chi(-r)
\overline{\chi_\lambda(g)}Use in source · 1 passage
Search aliases: overlineχ_λ(g) overline chi lambda
Use in source · Read passage · 02_strengthened.tex, line 207
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda = \frac{d_\lambda}{\abs{G}} \sum_{g\in G} \overline{\chi_\lambda(g)}U(g)
pUse in source · 21 passages
Search aliases: p
Use in source · monograph_atlas.tex, line 299
g Every effect in the computational basis E_a=ket abra a has the same expectation on both matrices, namely 1/2 at 00 and 11 and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture. headA measurement that resolves the difference Use the valid two-outcome measurement with effects E=ρ and I-E. Then Tr(Eρ)=1, Tr (E D_rm computational(ρ))=frac12. The effect itself is not diagonal in the fine computational packets. It comp…
Exact mathematical passage
\Tr(E\rho)=1,\qquad \Tr\!\left(E D_{\rm computational}(\rho)\right)=\frac12.
p^{\mathsf T}Use in source · 1 passage
Search aliases: p^ T mathsf
Use in source · Read passage · 03_supplement_05.tex, line 98
Rank-one rigidity of the corrected channel. The module tests the map equality on actual matrix units and flattens each A_ra into a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzero p, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication X…
Exact mathematical passage
X^*X=\overline p\,p^{\mathsf T}.
P^{(1)}_{j_1}Use in source · 2 passages
Search aliases: P^(1)_j_1
Use in source · Read passage · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
P^{(a)}_0Use in source · 1 passage
Search aliases: P^(a)_0
Use in source · Read passage · 01_finite_core.tex, line 238
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P^{(a)}_0,\ldots,P^{(a)}_{N-1}
P^{(a)}_{N-1}Use in source · 1 passage
Search aliases: P^(a)_N-1
Use in source · Read passage · 01_finite_core.tex, line 238
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P^{(a)}_0,\ldots,P^{(a)}_{N-1}
P^{(i)}_{a_i}Use in source · 1 passage
Search aliases: P^(i)_a_i
Use in source · Read passage · 03_supplement_02.tex, line 54
Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.
Exact mathematical passage
Q_a=\mathsf T\bigl(i\mapsto P^{(i)}_{a_i}\bigr), \qquad C_r=\sum_{\sum_i a_i=r}Q_a.
P^{(i)}_jUse in source · 2 passages
Search aliases: P^(i)_j
Use in source · Read passage · 03_supplement_02.tex, line 53
Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.
Exact mathematical passage
(P^{(i)}_j)_{j\in\mathbb Z/N\mathbb Z}
P^{(m)}_{j_m}Use in source · 2 passages
Search aliases: P^(m)_j_m
Use in source · Read passage · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
P^*Use in source · 4 passages
Search aliases: P^*
Use in source · Read passage · 01_finite_core.tex, line 792
Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environment can distinguish logical states and exact correction is impossible.
Exact mathematical passage
P=P^*=P^2
P_{\mathbf j}^{\mathrm{fine}}Use in source · 1 passage
Search aliases: P_ j^fine mathbf mathrm
Use in source · Read passage · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
P_{\bm j}^{\rm fine}Use in source · 1 passage
Search aliases: P_ j^rm fine bm rm
Use in source · Read passage · 02_strengthened.tex, line 343
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
P_{\bm j}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
P_{000}Use in source · 1 passage
Search aliases: P_000
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigpar…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_{011}Use in source · 1 passage
Search aliases: P_011
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigpar…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_1Use in source · 3 passages
Search aliases: P_1
Use in source · monograph_atlas.tex, line 555
vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILURETraditional edition: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5
Exact mathematical passage
P_1=I-P_0
P_{101}Use in source · 1 passage
Search aliases: P_101
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigpar…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_{110}Use in source · 1 passage
Search aliases: P_110
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigpar…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_2Use in source · 1 passage
Search aliases: P_2
Use in source · cyclic_four.tex, line 49
ll] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at (8.05,1.25) S v_3=- iv_3; node[font=small,align=center,font=small] at (11.85,1.…
Exact mathematical passage
P_2=v_2v_2^*
P_3Use in source · 1 passage
Search aliases: P_3
Use in source · cyclic_four.tex, line 51
at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at (8.05,1.25) S v_3=- iv_3; node[font=small,align=center,font=small] at (11.85,1.25) P_3=v_3v_3^*; node[font=small,align=center,font=scriptsize] at (8.8,0.12) Each column has four entries of magnitude 1/2.;
Exact mathematical passage
P_3=v_3v_3^*
p_aUse in source · 1 passage
Search aliases: p_a
Use in source · Read passage · 03_supplement_04.tex, line 126
Exact correction, including coherent errors. For every operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent sup…
Exact mathematical passage
\mathcal R\!\left(\sum_ap_aV_a\rho V_a^*\right)=\rho \quad\text{when}\quad\sum_ap_a=1.
P_a^*Use in source · 3 passages
Search aliases: P_a^*
Use in source · Read passage · 02_strengthened.tex, line 1013
The projectors P_a form a Kraus family because ∑_aP_a^*P_a=I. Thus the map is completely positive and trace preserving. It is unital for the same reason. Orthogonality and idempotence follow from [source reference]. Its range is the sum of the diagonal operator packets. This is exactly the commutant of the projector family.
Exact mathematical passage
\sum_aP_a^*P_a=I
P_a^2Use in source · 2 passages
Search aliases: P_a^2
Use in source · Read passage · 01_finite_core.tex, line 541
Differentiating P_a^2=P_a gives dot P_aP_a+P_adot P_a=dot P_a. Summing over a and using ∑_adot P_a=0 yields ∑_aP_adot P_a=-∑_adot P_aP_a, which proves both displayed formulas for C and its skew-Hermiticity.
Exact mathematical passage
P_a^2=P_a
P_{a_0}Use in source · 4 passages
Search aliases: P_a_0
Use in source · Read passage · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
P_{a_1}Use in source · 3 passages
Search aliases: P_a_1
Use in source · Read passage · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
P_{a_m}Use in source · 3 passages
Search aliases: P_a_m
Use in source · Read passage · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
P_{a_{m-1}}Use in source · 3 passages
Search aliases: P_a_m-1
Use in source · Read passage · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
P_b^2Use in source · 1 passage
Search aliases: P_b^2
Use in source · Read passage · 03_supplement_01.tex, line 41
UU^*=1 · Longrightarrow ∑_aU_ba(U_da)^*=δ_bdP_b. For the first law, expand each corner, collapse P_b^2 and sum: ∑_bP_aU^*P_bUP_c =P_aU^*(∑_bP_b)UP_c=P_aP_c. The second law follows by the analogous calculation and has its own formal proof. For complex matrices, the adjoint is the conjugate transpose. The order of the operator products is preserved throughout.
Exact mathematical passage
P_b^2
P_cUse in source · 5 passages
Search aliases: P_c
Use in source · monograph_atlas.tex, line 136
le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original 4×4 circuit matrix: ∑_a,b,cP_b CNOT P_a (H_rm Had⊗ I)P_c =CNOT(H_rm Had⊗ I). This expansion is an identity even though the intermediate parity …
Exact mathematical passage
\sum_{a,b,c}P_b\,\mathrm{CNOT}\,P_a (H_{\rm Had}\otimes I)P_c =\mathrm{CNOT}(H_{\rm Had}\otimes I).
P_c^2Use in source · 1 passage
Search aliases: P_c^2
Use in source · Read passage · 03_supplement_01.tex, line 16
0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.
Exact mathematical passage
P_c^2=P_c
P_dUse in source · 1 passage
Search aliases: P_d
Use in source · Read passage · 02_strengthened.tex, line 970
Idempotence and completeness follow from the projector identities. For Hilbert Schmidt orthogonality, ipP_aXP_bP_cYP_d_HS · = Tr(P_bX^*P_aP_cYP_d) · = δ_acδ_bd Tr(P_bX^*P_aYP_b). Since the packets are orthogonal and sum to the original operator, their squared norms add to its squared norm.
Exact mathematical passage
\begin{aligned} \ip{P_aXP_b}{P_cYP_d}_{\HS} &= \Tr(P_bX^*P_aP_cYP_d)\\ &= \delta_{ac}\delta_{bd} \Tr(P_bX^*P_aYP_b). \end{aligned}
P_{\mathrm{even}}Use in source · 1 passage
Search aliases: P_even mathrm
Use in source · Read passage · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
P_jUse in source · 22 passages
Search aliases: P_j
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
P_{j+q}Use in source · 3 passages
Search aliases: P_j+q
Use in source · Read passage · 02_strengthened.tex, line 306
[Charge selection rule] If A has charge q, then P_kAP_j=0 unless k=j+q in Z/N Z. In fact, AP_j=P_j+qAP_j.
Exact mathematical passage
AP_j=P_{j+q}AP_j.
P_j^{(a)}Use in source · 1 passage
Search aliases: P_j^(a)
Use in source · Read passage · 02_strengthened.tex, line 339
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
\bigl(P_j^{(a)}\bigr)_{j\in\ZZ/N\ZZ}.
P_j^*Use in source · 2 passages
Search aliases: P_j^*
Use in source · Read passage · 01_finite_core.tex, line 221
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
P_j^*=\frac1N\sum_{r=0}^{N-1}\omega^{jr}S^{-r}=P_j.
P_j^2Use in source · 1 passage
Search aliases: P_j^2
Use in source · monograph_atlas.tex, line 102
Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.
Exact mathematical passage
P_j^2=P_j
P_{j_1}^{(1)}Use in source · 2 passages
Search aliases: P_j_1^(1)
Use in source · Read passage · 02_strengthened.tex, line 343
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
P_{\bm j}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
P_{j_m}^{(m)}Use in source · 2 passages
Search aliases: P_j_m^(m)
Use in source · Read passage · 02_strengthened.tex, line 343
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
P_{\bm j}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
P_kUse in source · 5 passages
Search aliases: P_k
Use in source · monograph_atlas.tex, line 102
Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.
Exact mathematical passage
P_jP_k=0
p_kUse in source · 1 passage
Search aliases: p_k
Use in source · Read passage · 01_finite_core.tex, line 681
Let ρ be a density matrix and let {E_k} be a POVM: E_k≥0, ∑_kE_k=I. The outcome probabilities are p_k=Tr(E_kρ).
Exact mathematical passage
p_k=\Tr(E_k\rho).
P_mUse in source · 1 passage
Search aliases: P_m
Use in source · Read passage · 01_finite_core.tex, line 509
Let P_1(t),…,P_m(t) be differentiable on an open real time interval, with P_a(t)^*=P_a(t) and P_aP_b=δ_abP_a, ∑_aP_a=I.
Exact mathematical passage
P_1(t),\ldots,P_m(t)
P_nUse in source · 7 passages
Search aliases: P_n
Use in source · monograph_atlas.tex, line 795
nite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how finite quadratic forms pass positivity to the limit. xfigtrace_topology headOperator norm does not preserve trace in growing rank On ℓ^2, let P_n select the first n coordinates and set A_n=P_n/n. Then normA_n=1/n→0 while Tr A_n=1 for every n. These positive trace-class operat…
Exact mathematical passage
P_n
P_{\mathrm{odd}}Use in source · 1 passage
Search aliases: P_odd mathrm
Use in source · Read passage · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
P_{\phys}Use in source · 6 passages
Search aliases: P_phys phys mathrm
Use in source · Read passage · 02_strengthened.tex, line 45
Constraint closure needs a different obstruction. If C_a are self adjoint constraints and P_phys projects onto their common kernel, then P_phys[C_a,C_b]P_phys=0 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of [C_a,C_b] in the quotient of End( H) by the span of the constraints. Vanishing means that the constraint span is closed under commutators.
Exact mathematical passage
P_{\phys}
P_r^{\mathrm{coarse}}Use in source · 2 passages
Search aliases: P_r^coarse mathrm
Use in source · Read passage · 01_finite_core.tex, line 247
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
\boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. }
P_r^{\rm total}Use in source · 2 passages
Search aliases: P_r^rm total rm
Use in source · Read passage · 02_strengthened.tex, line 351
[Tensor charge refinement] For r∈ Z/N Z, define P_r^rm total = ∑_j_1+⋯+j_m=r P_j_1^(1)⊗⋯⊗ P_j_m^(m). Then the P_r^rm total form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.
Exact mathematical passage
P_r^{\rm total} = \sum_{j_1+\cdots+j_m=r} P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
P_{\rm even}Use in source · 2 passages
Search aliases: P_rm even rm
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigpar…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_{\rm odd}Use in source · 2 passages
Search aliases: P_rm odd rm
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigpar…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_{\alpha_0}Use in source · 1 passage
Search aliases: P_α_0 alpha
Use in source · Read passage · 01_finite_core.tex, line 283
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\boxed{ P_{\alpha_m}UP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}U_mP_{\alpha_{m-1}} \cdots P_{\alpha_1}U_1P_{\alpha_0}. }
P_{\alpha_1}Use in source · 1 passage
Search aliases: P_α_1 alpha
Use in source · Read passage · 01_finite_core.tex, line 283
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\boxed{ P_{\alpha_m}UP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}U_mP_{\alpha_{m-1}} \cdots P_{\alpha_1}U_1P_{\alpha_0}. }
P_{\alpha_m}Use in source · 1 passage
Search aliases: P_α_m alpha
Use in source · Read passage · 01_finite_core.tex, line 283
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\boxed{ P_{\alpha_m}UP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}U_mP_{\alpha_{m-1}} \cdots P_{\alpha_1}U_1P_{\alpha_0}. }
P_{\alpha_{m-1}}Use in source · 1 passage
Search aliases: P_α_m-1 alpha
Use in source · Read passage · 01_finite_core.tex, line 283
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\boxed{ P_{\alpha_m}UP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}U_mP_{\alpha_{m-1}} \cdots P_{\alpha_1}U_1P_{\alpha_0}. }
P_\gammaUse in source · 2 passages
Search aliases: P_γ gamma
Use in source · Read passage · 01_finite_core.tex, line 319
Using completeness and unitarity, ∑_β U_βα^*U_βγ = P_α U^*(∑_β P_β)UP_γ = P_α P_γ. For the second identity, sum over the source packets and use unitarity in the reverse order.
Exact mathematical passage
\sum_\beta U_{\beta\alpha}^*U_{\beta\gamma} = P_\alpha U^*\left(\sum_\beta P_\beta\right)UP_\gamma = P_\alpha P_\gamma.
P_\deltaUse in source · 1 passage
Search aliases: P_δ delta
Use in source · Read passage · 01_finite_core.tex, line 751
Packet superoperator blocks are defined by Φ_αβ;γδ(X) = P_α Φ(P_γ XP_δ) P_β. Summing all four packet indices reconstructs Φ(X) exactly.
Exact mathematical passage
\Phi_{\alpha\beta;\gamma\delta}(X) = P_\alpha \Phi(P_\gamma XP_\delta) P_\beta.
P_\lambdaUse in source · 5 passages
Search aliases: P_λ lambda
Use in source · monograph_atlas.tex, line 534
, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and three-cycle classes are c|rrr|r · e · (12) · (123) · d_λ trivial · 1 · 1 · 1 · 1 sign · 1 · -1 · 1 · 1 standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3…
Exact mathematical passage
\begin{array}{c|rrr|r} & e & (12)&(123)&d_\lambda\\\hline \text{trivial}&1&1&1&1\\ \text{sign}&1&-1&1&1\\ \text{standard}&2&0&-1&2 \end{array} \qquad P_\lambda=\frac{d_\lambda}{6}\sum_{g\in S_3}\chi_\lambda(g^{-1})U(g).
P_\muUse in source · 1 passage
Search aliases: P_μ mu
Use in source · Read passage · 02_strengthened.tex, line 215
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda, \qquad \sum_{\lambda\in\widehat G}P_\lambda=I_{\Hcal}.
P_{\ell+1,a}Use in source · 1 passage
Search aliases: P_ℓ+1,a ell
Use in source · Read passage · 02_strengthened.tex, line 2404
[Exact packet direct limit] Assume the packet families have one fixed finite address set at every level and each packet address a is compatible in the sense P_ℓ+1,aR_ℓ=R_ℓ P_ℓ,a. Then there is an orthogonal projector P_∞,a on H_∞ satisfying P_∞,aι_ℓ=ι_ℓ P_ℓ,a. Finite completeness and orthogonality pass to the direct limit.
Exact mathematical passage
P_{\ell+1,a}R_\ell=R_\ell P_{\ell,a}.
P_{\ell+1,\beta}Use in source · 1 passage
Search aliases: P_ℓ+1,β ell beta
Use in source · Read passage · 01_finite_core.tex, line 1026
A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Exact mathematical passage
\boxed{ P_{\ell,\alpha} = C_\ell \left( \sum_{\beta\succ\alpha}P_{\ell+1,\beta} \right) E_\ell. }
P_{\infty,a}Use in source · 2 passages
Search aliases: P_∞,a infty
Use in source · Read passage · 02_strengthened.tex, line 2407
[Exact packet direct limit] Assume the packet families have one fixed finite address set at every level and each packet address a is compatible in the sense P_ℓ+1,aR_ℓ=R_ℓ P_ℓ,a. Then there is an orthogonal projector P_∞,a on H_∞ satisfying P_∞,aι_ℓ=ι_ℓ P_ℓ,a. Finite completeness and orthogonality pass to the direct limit.
Exact mathematical passage
P_{\infty,a}
\mathrm{phys}Use in source · 5 passages
Search aliases: phys mathrm
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\physUse in source · 10 passages
Search aliases: phys mathrm
Use in source · Read passage · 02_strengthened.tex, line 45
Constraint closure needs a different obstruction. If C_a are self adjoint constraints and P_phys projects onto their common kernel, then P_phys[C_a,C_b]P_phys=0 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of [C_a,C_b] in the quotient of End( H) by the span of the constraints. Vanishing means that the constraint span is closed under commutators.
Exact mathematical passage
P_{\phys}
\pmodUse in source · 1 passage
Search aliases: pmod
Use in source · Read passage · 01_finite_core.tex, line 247
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
\boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. }
\PrUse in source · 1 passage
Search aliases: Pr
Use in source · monograph_atlas.tex, line 198
in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε t=π/6, the amplitude is 1/2 and the probability is 1/4. Near zero, amplitudes are first order while escape probabilities are second order. Mixing thos…
Exact mathematical passage
\Pr(\text{hidden packet at }t)\le\min\{1,\varepsilon^2t^2\}.
Q^2Use in source · 1 passage
Search aliases: Q^2
Use in source · Read passage · 03_supplement_04.tex, line 107
Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.
Exact mathematical passage
Q^2=Q=Q^*
Q^\perpUse in source · 2 passages
Search aliases: Q^⊥ perp
Use in source · Read passage · 02_strengthened.tex, line 1385
The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.
Exact mathematical passage
Q^\perp=I-Q
q_1Use in source · 3 passages
Search aliases: q_1
Use in source · Read passage · 02_strengthened.tex, line 369
[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.
Exact mathematical passage
q_1+\cdots+q_m
Q_aUse in source · 12 passages
Search aliases: Q_a
Use in source · monograph_atlas.tex, line 349
vpage19Find the error without reading the logical stateFOUR EXACT SYNDROME PACKETSTraditional edition: Theorem II.8.2, Theorem II.8.4, Example II.8.8. Encode a qubit by ket0_L=ket000 and ket1_L=ket111. Let P project onto this code and take errors E_a∈{I,X_1,X_2,X_3}. Their images are four orthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*…
Exact mathematical passage
Q_a=E_aPE_a^*
q_aUse in source · 1 passage
Search aliases: q_a
Use in source · Read passage · 02_strengthened.tex, line 365
[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.
Exact mathematical passage
q_a
Q_a^*Use in source · 1 passage
Search aliases: Q_a^*
Use in source · Read passage · 03_supplement_02.tex, line 74
Expanding the finite product of sums gives T(A) T(B)= T(i↦ A_iB_i). Consequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so Q_a^*=Q_a. Summing over all address functions and using the same expansion yields ∑_a Q_a= T(i↦∑_jP^(i)_j)=I. A separate algebraic regrouping theorem applies to any finite orthogonal family and a…
Exact mathematical passage
Q_a^*=Q_a
Q_cUse in source · 1 passage
Search aliases: Q_c
Use in source · Read passage · 02_strengthened.tex, line 1042
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
(Q_c)
q_FUse in source · 2 passages
Search aliases: q_F
Use in source · Read passage · 02_strengthened.tex, line 1295
Exact correction of an error span means that one completely positive trace preserving recovery R satisfies R(Fρ F^*)=q_Fρ for every F in that span and every code-supported matrix ρ, with a nonnegative scalar q_F depending on F but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restores the code input with total scalar one. Individual unnormalized errors are not required to have unit recovery weight.
Exact mathematical passage
\Rcal(F\rho F^*)=q_F\rho
Q_jUse in source · 7 passages
Search aliases: Q_j
Use in source · Read passage · 02_strengthened.tex, line 1331
the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
Q_j=V_jV_j^*
q_mUse in source · 3 passages
Search aliases: q_m
Use in source · Read passage · 02_strengthened.tex, line 369
[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.
Exact mathematical passage
q_1+\cdots+q_m
q_nUse in source · 4 passages
Search aliases: q_n
Use in source · monograph_atlas.tex, line 788
vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPETraditional edition: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how …
Exact mathematical passage
q_n=1/2+2^{-(n+2)}
q_n^{j+k}Use in source · 1 passage
Search aliases: q_n^j+k
Use in source · monograph_atlas.tex, line 788
vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPETraditional edition: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how …
Exact mathematical passage
C_n(j,k)=1+q_n^{j+k}
Q_rUse in source · 3 passages
Search aliases: Q_r
Use in source · Read passage · 02_strengthened.tex, line 1740
[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.
Exact mathematical passage
D=\sum_{r}E_rQ_r
q_rUse in source · 3 passages
Search aliases: q_r
Use in source · Read passage · 02_strengthened.tex, line 1236
[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.
Exact mathematical passage
q_r
\mathfrak Q_\ellUse in source · 1 passage
Search aliases: Q_ℓ mathfrak ell
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\QcalUse in source · 6 passages
Search aliases: Qcal Q mathcal
Use in source · Read passage · 02_strengthened.tex, line 759
Let (P_a) and (Q_a) be two complete packet systems with the same index set. Define S( Q, P)=∑_aQ_aP_a.
Exact mathematical passage
S(\Qcal,\Pcal)=\sum_aQ_aP_a.
\Qcal_{cd}Use in source · 1 passage
Search aliases: Qcal_cd Q_cd Qcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 1043
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
\Phi_{cd,ab} = \Qcal_{cd}\Phi\Pcal_{ab}.
\QQUse in source · 2 passages
Search aliases: QQ
Use in source · monograph_atlas.tex, line 692
egative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i), so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positiv…
Exact mathematical passage
\QQ(\ii)
\RUse in source · 5 passages
Search aliases: R mathbb
Use in source · Read passage · 03_supplement_01.tex, line 150
Checked stronger statement. For every k,h∈ R with h≠0, 0≤ k^2-λ_k,h≤frack^4h^212. The code proves this without the low-mode restriction. The bound may be loose away from the low-mode regime, but it remains valid.
Exact mathematical passage
k,h\in\R
\mathbb RUse in source · 1 passage
Search aliases: R mathbb
Use in source · Read passage · 04_alternative_derivations.tex, line 5
ses and notation agree with the corrected finite core. The earlier abbreviated argument for error correction is replaced by the complete normalized and scaled recovery proofs above, and its substantive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t …
Exact mathematical passage
t\in\mathbb R
R'Use in source · 1 passage
Search aliases: R'
Use in source · Read passage · 02_strengthened.tex, line 2145
By [source reference], Z_d(t) =(4π t)^-d/2(1+R(t)), where R(t)=∑_n≠0 e^-absn^2/(4t) =O( e^-1/(4t)). Differentiation gives tR'(t)=O(t^-1 e^-1/(4t)). Substituting this estimate into the logarithmic derivative gives the stated small time expansion.
Exact mathematical passage
tR'(t)=O\left(t^{-1}\e^{-1/(4t)}\right).
R^*Use in source · 1 passage
Search aliases: R^*
Use in source · monograph_atlas.tex, line 556
vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILURETraditional edition: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5 4/5 · 3/5, Q_a=RP_aR^*, S=∑_a=0^1Q_aP_a=frac35R. The exact Gram matrices give (S-I)^*(S-I)=frac1625I, S^*S=frac925I, normS-I=frac45<1. Consequently the unitary polar factor is S(S^*S)^-1/2=R, and Q_aR=RP_a for both complete packets. xfigpolar headThe norm condition selects a chart If the target rank-one…
Exact mathematical passage
R=\mat{3/5&-4/5\\4/5&3/5},\qquad Q_a=RP_aR^*, \qquad S=\sum_{a=0}^1Q_aP_a=\frac35R.
R^\nablaUse in source · 2 passages
Search aliases: R^∇ nabla
Use in source · Read passage · 03_supplement_06.tex, line 125
Cancellation of the second derivatives. The product rule and the identity for the ambient Lie bracket reduce curvature to R^∇(X,Y)s=P d P[X] d s[Y]-P d P[Y] d s[X]. Substitute the identity obtained by differentiating the range condition for the section. The terms containing P d P[v]P vanish, leaving R^∇(X,Y)s= P( d P[X] d P[Y]- d P[Y] d P[X])P s. Lean also proves that this depends only on X(x), Y(x) and s(x), not on their extensions. The operator-valued form is alternating, additive and homogene…
Exact mathematical passage
R^\nabla(X,Y)s=P\dd P[X]\dd s[Y]-P\dd P[Y]\dd s[X].
R_1Use in source · 2 passages
Search aliases: R_1
Use in source · monograph_atlas.tex, line 570
Short steps can be composed before reaching a distant endpoint. For example, the rational step R_1=mat4/5 · -3/5 3/5 · 4/5 obeys R_1^2=mat7/25 · -24/25
Exact mathematical passage
R_1=\mat{4/5&-3/5\\3/5&4/5}
R_1^*Use in source · 1 passage
Search aliases: R_1^*
Use in source · monograph_atlas.tex, line 613
The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery returns every code matrix unit, including both off-diagonal units, exactly. It need not preserve states on the unused complement. headExceptional cases and …
Exact mathematical passage
R_1=P,\qquad R_2=\ket0\bra2, \qquad R_1^*R_1+R_2^*R_2=I_3.
R_1^2Use in source · 1 passage
Search aliases: R_1^2
Use in source · monograph_atlas.tex, line 571
3/5 · 4/5 obeys R_1^2=mat7/25 · -24/25 24/25 · 7/25. The multiplication is exact and each step has a valid local overlap. The convergence theorem has a separate analytic contract: a twice continuously differentiable complete projector path on a compact time interval, sufficiently fine partitions, and a constant depending on the path in an O(|π|) estimate in operator norm. headEvidence boundary The retained run checks the Gram identities, polar factor, intertwining, and product of two rational st…
Exact mathematical passage
R_1^2=\mat{7/25&-24/25\\24/25&7/25}.
R_2Use in source · 1 passage
Search aliases: R_2
Use in source · monograph_atlas.tex, line 613
The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery returns every code matrix unit, including both off-diagonal units, exactly. It need not preserve states on the unused complement. headExceptional cases and …
Exact mathematical passage
R_1=P,\qquad R_2=\ket0\bra2, \qquad R_1^*R_1+R_2^*R_2=I_3.
R_2^*Use in source · 1 passage
Search aliases: R_2^*
Use in source · monograph_atlas.tex, line 613
The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery returns every code matrix unit, including both off-diagonal units, exactly. It need not preserve states on the unused complement. headExceptional cases and …
Exact mathematical passage
R_1=P,\qquad R_2=\ket0\bra2, \qquad R_1^*R_1+R_2^*R_2=I_3.
R_aUse in source · 6 passages
Search aliases: R_a
Use in source · monograph_atlas.tex, line 353
rthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an actual channel The operators R_a=E_a^*Q_a satisfy ∑_aR_a^*R_a=I. Applying the recovery R(X)=∑_aR_aXR_a^* to the noisy state gives exactly ψψ^*. The off-diagonal logic…
Exact mathematical passage
R_a=E_a^*Q_a
R_a^*Use in source · 3 passages
Search aliases: R_a^*
Use in source · monograph_atlas.tex, line 353
ets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an actual channel The operators R_a=E_a^*Q_a satisfy ∑_aR_a^*R_a=I. Applying the recovery R(X)=∑_aR_aXR_a^* to the noisy state gives exactly ψψ^*. The off-diagonal logical coherence 12/25 is r…
Exact mathematical passage
\sum_aR_a^*R_a=I
R_{\mathrm{extra}}Use in source · 2 passages
Search aliases: R_extra mathrm
Use in source · Read passage · 03_supplement_04.tex, line 109
The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family I_J⊗ R_a, including the complement term. Positivity is …
Exact mathematical passage
R_{\mathrm{extra}}=Q
R_{\mathrm{extra}}^*Use in source · 1 passage
Search aliases: R_extra^* mathrm
Use in source · Read passage · 03_supplement_04.tex, line 114
The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family I_J⊗ R_a, including the complement term. Positivity is …
Exact mathematical passage
\sum_aR_a^*R_a+R_{\mathrm{extra}}^*R_{\mathrm{extra}} =\sum_aV_aV_a^*+Q=S+Q=I.
R_nUse in source · 2 passages
Search aliases: R_n
Use in source · monograph_atlas.tex, line 769
vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and…
Exact mathematical passage
H_{n+1}R_n=R_nH_n
R_PUse in source · 3 passages
Search aliases: R_P
Use in source · Read passage · 03_supplement_04.tex, line 70
Retaining the bracket of vector fields. Define the actual curvature action by R_P(X,Y)s=∇_X∇_Ys-∇_Y∇_Xs-∇_[X,Y]s. The source uses the Lie bracket defined through Fr echet derivatives. Symmetry of second derivatives is supplied by the C^2 hypothesis, not postulated separately. Expanding the connection gives R_P(X,Y)s=P(DP[X]Ds[Y]-DP[Y]Ds[X]). Substituting the section identity and removing the zero sandwich terms proves R_P(X,Y)s=P(DP[X]DP[Y]-DP[Y]DP[X])P s. Thus the coefficient is precisely P( d …
Exact mathematical passage
R_P(X,Y)s=\nabla_X\nabla_Ys-\nabla_Y\nabla_Xs-\nabla_{[X,Y]}s.
R_rUse in source · 5 passages
Search aliases: R_r
Use in source · Read passage · 03_supplement_05.tex, line 10
recise finite channel statement Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^*R_r=I. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire ambien…
Exact mathematical passage
\mathcal R(\sigma)=\sum_r R_r\sigma R_r^*
R_r^*Use in source · 3 passages
Search aliases: R_r^*
Use in source · Read passage · 03_supplement_05.tex, line 10
recise finite channel statement Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^*R_r=I. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire ambien…
Exact mathematical passage
\mathcal R(\sigma)=\sum_r R_r\sigma R_r^*
\RanUse in source · 6 passages
Search aliases: Ran operatorname
Use in source · Read passage · 01_finite_core.tex, line 598
textitRange coordinates. Fix a∈ M and put T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). Then T_a(a)=I and P(x)T_a(x)=T_a(x)P(a). Invertibility is open, so on a neighborhood of a, T_a(x) restricts to an isomorphism from Ran P(a) to Ran P(x). This proves local constancy of rank. The local coordinates and their transitions are (x,v)⟼ (x,T_a(x)^-1v), u⟼ T_b(x)^-1T_a(x)u. Their smoothness follows from smoothness of P and inversion. The transition maps satisfy the cocycle identity by cancellation. These charts co…
Exact mathematical passage
\Ran P(a)
\Rcal_0Use in source · 4 passages
Search aliases: Rcal_0 R_0 Rcal mathcal
Use in source · Read passage · 02_strengthened.tex, line 1361
[Exact recovery on the error span] Under the hypotheses of [source reference], define on the syndrome support Q=∑_λ_j>0Q_j the completely positive map R_0(X) = ∑_λ_j>0 V_j^*Q_jXQ_jV_j. It extends to a quantum channel R on End( H). If Φ(X)=∑_aE_aXE_a^* is trace preserving on code states, then RΦ(ρ)=ρ for every density matrix ρ=Pρ P.
Exact mathematical passage
\Rcal_0(X) = \sum_{\lambda_j>0} V_j^*Q_jXQ_jV_j.
\ReUse in source · 1 passage
Search aliases: Re
Use in source · monograph_atlas.tex, line 156
ity. Starting from ket0, there are two resolved paths to output ket1. Their signed products are bra1H_rm Hadket0bra0H_rm Hadket0=frac12, bra1H_rm Hadket1bra1H_rm Hadket0=-frac12. Their sum is zero. Squaring each separately gives a different answer. vfiginterference headWhere the missing term goes For path amplitudes a,b, the outcome probability is |a+b|^2=|a|^2+|b|^2+2Re(abar b). Here the last term is -1/2, canceling the first two terms. The path picture retains this information only if its edge…
Exact mathematical passage
|a+b|^2=|a|^2+|b|^2+2\Re(a\bar b)
\operatorname{Re}Use in source · 2 passages
Search aliases: Re operatorname
Use in source · Read passage · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\operatorname{Row}Use in source · 2 passages
Search aliases: Row operatorname
Use in source · Read passage · 03_supplement_03.tex, line 8
A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d× m coefficient space is equivalent to rank C=m. Consequently a cyclic seed requires m≤ d. The assumption that there is at least one row is essential …
Exact mathematical passage
\operatorname{Row}(C)
S^{-r}Use in source · 2 passages
Search aliases: S^-r
Use in source · Read passage · 01_finite_core.tex, line 221
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
P_j^*=\frac1N\sum_{r=0}^{N-1}\omega^{jr}S^{-r}=P_j.
S^2Use in source · 4 passages
Search aliases: S^2
Use in source · monograph_atlas.tex, line 494
age27One calculation of transfer to the opposite siteSEEDS, CHARACTERS, INTERFERENCE, AND DEPHASINGTraditional edition: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4. Return to the shift on four sites S and Laplacian L=2I-S-S^*. The same character projectors have energies (0,2,4,2). At time t=π/2, their evolution phases are (1,-1,1,-1), exactly the character eigenvalues of S^2. Hence e^- iπ L/2=S^2, ket0⟼ket2. vfigperfect_transfer At halfway time π/4, the actual amplitude vector is (- i,1, i,…
Exact mathematical passage
S^2
S^NUse in source · 6 passages
Search aliases: S^N
Use in source · Read passage · 01_finite_core.tex, line 189
Let N≥1 be an integer, and let S be a unitary symmetry whose order divides N: S^N=I. Set ω=e^2π i/N.
Exact mathematical passage
S^N=I.
S^rUse in source · 4 passages
Search aliases: S^r
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
S^{r+s}Use in source · 1 passage
Search aliases: S^r+s
Use in source · Read passage · 01_finite_core.tex, line 216
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
S^{r+s}=S^{r+s\bmod N}
S^{r+s\bmod N}Use in source · 1 passage
Search aliases: S^r+sbmod N bmod
Use in source · Read passage · 01_finite_core.tex, line 216
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
S^{r+s}=S^{r+s\bmod N}
S^{\operatorname{val}(r)}Use in source · 1 passage
Search aliases: S^val(r) operatorname
Use in source · Read passage · 03_supplement_03.tex, line 98
003 Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N is proved from S^N=I, giving ρ(r+s)=ρ(r)ρ(s) and ρ(0)=I. The final matrix theorem derives the representation law and character orthogonality from the …
Exact mathematical passage
\rho(r)=S^{\operatorname{val}(r)}.
S_3Use in source · 4 passages
Search aliases: S_3
Use in source · monograph_atlas.tex, line 533
imitation of testing only the common kernel on pagerefV26. takeawayExact finite structure is valuable because its boundary is explicit. The appendix adds understanding and reproducibility, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible character…
Exact mathematical passage
\HH=\CC[S_3]
S_aUse in source · 2 passages
Search aliases: S_a
Use in source · Read passage · 03_supplement_04.tex, line 104
Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.
Exact mathematical passage
S_a=V_aV_a^*,\qquad S=\sum_aS_a,\qquad Q=1-S.
S_bUse in source · 1 passage
Search aliases: S_b
Use in source · Read passage · 03_supplement_04.tex, line 107
Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.
Exact mathematical passage
S_aS_b=\delta_{ab}S_a
S_zUse in source · 5 passages
Search aliases: S_z
Use in source · Read passage · 03_supplement_01.tex, line 106
-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).
Exact mathematical passage
D_z=zI_Q-D,\qquad S_z=zI_P-A-BD_z^{-1}C.
S_z^{-1}Use in source · 1 passage
Search aliases: S_z^-1
Use in source · Read passage · 03_supplement_01.tex, line 114
Correct statement: source Theorem 9.1. Assume both D_z and zI-H are invertible. Then S_z is invertible and ((zI-H)^-1)_PP=S_z^-1. The left side is the submatrix of the full inverse on the retained coordinates. This matches the paper's compressed operator when written in an orthonormal basis of the retained subspace. The proof is valid for arbitrary complex blocks of compatible sizes and does not require Hermiticity.
Exact mathematical passage
\boxed{\bigl((zI-H)^{-1}\bigr)_{PP}=S_z^{-1}.}
\sinUse in source · 45 passages
Search aliases: sin
Use in source · monograph_atlas.tex, line 188
vpage11Quantify the amplitude that escapesCOMMUTATORS AND PROBABILITY BOUNDSTraditional edition: Theorem II.5.3, Corollary II.5.4. For a two-level system, choose H=ε X, P=ket0bra0, and Q=I-P. Direct exponentiation gives e^- i tH=cos(ε t)I- isin(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probabi…
Exact mathematical passage
e^{-\ii tH}=\cos(\varepsilon t)I-\ii\sin(\varepsilon t)X, \quad \norm{Qe^{-\ii tH}P}=|\sin(\varepsilon t)|, \quad\norm{[H,P]}=|\varepsilon|.
\sin^2Use in source · 16 passages
Search aliases: sin^2 sin
Use in source · monograph_atlas.tex, line 197
irect exponentiation gives e^- i tH=cos(ε t)I- isin(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε t=π/6, the amplitude is 1/2 and the probability is 1/4. Near zero, amplitudes are first order while e…
Exact mathematical passage
\sin^2(\varepsilon t)
\SpanUse in source · 2 passages
Search aliases: Span
Use in source · monograph_atlas.tex, line 61
ment changes the local spanSCHMIDT RANK AS ACCESSIBLE MULTIPLICITYTraditional edition: Theorem II.2.2, Corollary II.2.4. Write a bipartite vector as a coefficient matrix C, with the first subsystem indexing rows. Acting only on that subsystem transforms C to AC. Every resulting row remains in the row space of C, and every independent choice of such rows can be obtained in this way. dimSpan{(A⊗ I)ψ:A∈ M_d( C)} =d rank C. vfiglocal_orbits The product vector ket00 has matrix C=diag(1,0). Its local …
Exact mathematical passage
\dim\Span\{(A\otimes I)\psi:A\in M_d(\CC)\} =d\,\rank C.
\operatorname{span}_{\CC}Use in source · 1 passage
Search aliases: span_CC span_ C operatorname CC operatorname mathbb
Use in source · Read passage · 02_strengthened.tex, line 2175
Let C_1,…,C_m be self adjoint constraint operators on a finite dimensional Hilbert space. The distinction between constraint action and constraint closure is standard in constrained Hamiltonian theory [source reference]. Write c=span_ C{C_1,…,C_m} ⊆End( H).
Exact mathematical passage
\mathfrak c=\operatorname{span}_{\CC}\{C_1,\ldots,C_m\} \subseteq\End(\Hcal).
\spanofUse in source · 1 passage
Search aliases: spanof span operatorname
Use in source · Read passage · 03_supplement_03.tex, line 63
The matrix-orbit dimension and the coefficient equivalence now yield dim_ Cspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r(ψ).
Exact mathematical passage
\boxed{\dim_\C\spanof\{(A\otimes I_B)\psi:A\in M_{d_A}(\C)\} =d_A r(\psi).}
\Spec_\RUse in source · 2 passages
Search aliases: Spec_R Spec_ R Spec R operatorname mathbb
Use in source · Read passage · 03_supplement_03.tex, line 147
tual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator that was actually defined: Spec_ R T_e(t)={e^-t(x-e):x∈Spec_ R H}. Hermiticity also identifies the full complex spectrum with the image of these real e…
Exact mathematical passage
\Spec_\R T_e(t)=\{e^{-t(x-e)}:x\in\Spec_\R H\}.
\operatorname{srank}Use in source · 3 passages
Search aliases: srank operatorname
Use in source · monograph_atlas.tex, line 83
dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines the required count: srank_AA(HH)=max_ilceil m_i/d_irceil. This is an invariant of the declared representation of the algebra, and depends on more tha…
Exact mathematical passage
\operatorname{srank}_{\AA}(\HH)=\max_i\lceil m_i/d_i\rceil.
\operatorname{srank}_{\AA}Use in source · 1 passage
Search aliases: srank_AA operatorname AA
Use in source · monograph_atlas.tex, line 83
dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines the required count: srank_AA(HH)=max_ilceil m_i/d_irceil. This is an invariant of the declared representation of the algebra, and depends on more tha…
Exact mathematical passage
\operatorname{srank}_{\AA}(\HH)=\max_i\lceil m_i/d_i\rceil.
\operatorname{srank}_{\Acal}Use in source · 1 passage
Search aliases: srank_Acal srank_ A operatorname Acal operatorname mathcal
Use in source · Read passage · 02_strengthened.tex, line 135
[Quantum seed rank] For the displayed multiplicity decomposition, srank_ A( H) = max_1≤ r≤ s lceilfracm_rd_rrceil.
Exact mathematical passage
\operatorname{srank}_{\Acal}(\Hcal) = \max_{1\le r\le s} \left\lceil\frac{m_r}{d_r}\right\rceil.
\operatorname{srank}_{\cA}Use in source · 1 passage
Search aliases: srank_cA srank_ A operatorname cA operatorname mathcal
Use in source · Read passage · 01_finite_core.tex, line 117
[Quantum seed-rank formula] For A=⊕_i(M_d_i( C)⊗ I_m_i) on H=⊕_i( C^d_i⊗ C^m_i), the minimum number of seeds required to generate H is srank_ A( H) = max_ilceilfracm_id_irceil.
Exact mathematical passage
\boxed{ \operatorname{srank}_{\cA}(\cH) = \max_i\left\lceil\frac{m_i}{d_i}\right\rceil. }
\succUse in source · 1 passage
Search aliases: succ
Use in source · Read passage · 01_finite_core.tex, line 1026
A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Exact mathematical passage
\boxed{ P_{\ell,\alpha} = C_\ell \left( \sum_{\beta\succ\alpha}P_{\ell+1,\beta} \right) E_\ell. }
\succeqUse in source · 4 passages
Search aliases: succeq
Use in source · Read passage · 03_supplement_02.tex, line 203
Matrix multiplication distributes across addition and scalar multiplication, which gives complex linearity. If ρsucceq0, each term K_aρ K_a^* is positive semidefinite, hence so is their finite sum.
Exact mathematical passage
\rho\succeq0
\supUse in source · 4 passages
Search aliases: sup
Use in source · Read passage · 02_strengthened.tex, line 2372
[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\sup_\ell\norm{H_\ell}<\infty.
\sup_{0\leq t\leq T}Use in source · 1 passage
Search aliases: sup_0≤ t≤ T sup leq leq
Use in source · Read passage · 02_strengthened.tex, line 2435
Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{\e^{-tH_n}-\e^{-tH}} \leq T\norm{H_n-H}.
\sup_{\abs t\leq T}Use in source · 2 passages
Search aliases: sup_abs t≤ T sup abs leq
Use in source · Read passage · 02_strengthened.tex, line 2428
[Norm summable Hamiltonian tower] Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{\abs t\leq T} \norm{\e^{-\ii tH_n}-\e^{-\ii tH}} \leq T\norm{H_n-H}.
\sup_\ellUse in source · 1 passage
Search aliases: sup_ℓ sup ell
Use in source · Read passage · 02_strengthened.tex, line 2372
[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\sup_\ell\norm{H_\ell}<\infty.
\tUse in source · 2 passages
Search aliases: t
Use in source · defect.tex, line 4
draw[->] (1,1.6)--(11,1.6);foreach x/t in 3/9/10,6/1,9/11/10 fill (x,1.6) circle (0.08);node[below] at (x,1.5) t; draw[<->] (3,2.5)--(6,2.5);draw[<->] (6,2.5)--(9,2.5); node[above] at (4.5,2.5) 1/10;node[above] at (7.5,2.5) 1/10; node[font=small] at (6,0.45) one pair: 1/10, full family {I,A}: 1/5;
Exact mathematical passage
\t
t^{-1}Use in source · 2 passages
Search aliases: t^-1
Use in source · Read passage · 02_strengthened.tex, line 2125
[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.
Exact mathematical passage
d_{\rm torus}(t) = d+O\left(t^{-1}\e^{-1/(4t)}\right)
t^2Use in source · 2 passages
Search aliases: t^2
Use in source · monograph_atlas.tex, line 198
in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε t=π/6, the amplitude is 1/2 and the probability is 1/4. Near zero, amplitudes are first order while escape probabilities are second order. Mixing thos…
Exact mathematical passage
\Pr(\text{hidden packet at }t)\le\min\{1,\varepsilon^2t^2\}.
T^{n_j}Use in source · 2 passages
Search aliases: T^n_j
Use in source · monograph_atlas.tex, line 717
tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This is a finite Gram factorization. The retained run checks the integer sem…
Exact mathematical passage
\sum_{j,k}\langle u_j,T^{n_j+n_k}u_k\rangle =\left\|\sum_jT^{n_j}u_j\right\|^2.
T^{n_j+n_k}Use in source · 5 passages
Search aliases: T^n_j+n_k
Use in source · monograph_atlas.tex, line 717
tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This is a finite Gram factorization. The retained run checks the integer sem…
Exact mathematical passage
\sum_{j,k}\langle u_j,T^{n_j+n_k}u_k\rangle =\left\|\sum_jT^{n_j}u_j\right\|^2.
T^{n_k}Use in source · 2 passages
Search aliases: T^n_k
Use in source · Read passage · 02_strengthened.tex, line 1890
Since T is positive and self adjoint, ipv_jT^n_j+n_kv_k = ipT^n_jv_jT^n_kv_k. The quadratic form is the squared norm of ∑_kc_kT^n_kv_k.
Exact mathematical passage
\ip{v_j}{T^{n_j+n_k}v_k} = \ip{T^{n_j}v_j}{T^{n_k}v_k}.
T_0Use in source · 1 passage
Search aliases: T_0
Use in source · monograph_atlas.tex, line 261
E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). It equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates. vfigrange_atlas headAn exact example of the domain restriction For the rotating line and anchor P(0), T_0(t)=cos t U(t), det T_0(t)=cos^2t. The chart fails at t=π/2, but P(π/2)=diag(0,1) is a perfectly regular rank-one proje…
Exact mathematical passage
T_0(t)=\cos t\,U(t),\qquad \det T_0(t)=\cos^2t.
t_0Use in source · 2 passages
Search aliases: t_0
Use in source · Read passage · 02_strengthened.tex, line 800
[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
0=t_0<t_1<\cdots<t_n=T,
t_1Use in source · 3 passages
Search aliases: t_1
Use in source · Read passage · 02_strengthened.tex, line 800
[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
0=t_0<t_1<\cdots<t_n=T,
t_aUse in source · 3 passages
Search aliases: t_a
Use in source · monograph_atlas.tex, line 430
ens also to have nonnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent…
Exact mathematical passage
t_a\ge0
T_{a+b}Use in source · 1 passage
Search aliases: T_a+b
Use in source · Read passage · 03_supplement_02.tex, line 29
Composition gives T_aT_b=T_a+b, and direct evaluation gives T_aket b=keta+b. Every binary address satisfies a+a=0, so each translation is an involution. In particular, the local operator changes precisely the addressed bit: X_jket a=keta with coordinate j replaced by a_j+1. The selected factor at j equals translation by the address supported at j with value a_j. Induction on a list proves that the product of translations is translation by their sum. Each of these addresses contributes one coordi…
Exact mathematical passage
T_aT_b=T_{a+b}
T_bUse in source · 2 passages
Search aliases: T_b
Use in source · Read passage · 01_finite_core.tex, line 599
textitRange coordinates. Fix a∈ M and put T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). Then T_a(a)=I and P(x)T_a(x)=T_a(x)P(a). Invertibility is open, so on a neighborhood of a, T_a(x) restricts to an isomorphism from Ran P(a) to Ran P(x). This proves local constancy of rank. The local coordinates and their transitions are (x,v)⟼ (x,T_a(x)^-1v), u⟼ T_b(x)^-1T_a(x)u. Their smoothness follows from smoothness of P and inversion. The transition maps satisfy the cocycle identity by cancellation. These charts co…
Exact mathematical passage
(x,v)\longmapsto (x,T_a(x)^{-1}v),\qquad u\longmapsto T_b(x)^{-1}T_a(x)u.
t_bUse in source · 1 passage
Search aliases: t_b
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
T_eUse in source · 5 passages
Search aliases: T_e
Use in source · Read passage · 03_supplement_03.tex, line 136
The actual Euclidean transfer exponential. Let H be a Hermitian matrix on a nonzero complex state space of finite dimension. The source defines T_e(t)=exp(-t(H-eI)) using Lean's exponential defined by a power series. The proof then relates this defined operator to its spectral expansion.
Exact mathematical passage
T_e(t)=\exp\bigl(-t(H-eI)\bigr)
T_{E_0}Use in source · 1 passage
Search aliases: T_E_0
Use in source · Read passage · 03_supplement_03.tex, line 161
Positivity, semigroup composition, and the interval. Every eigenvalue of T_e(t) is strictly positive, for arbitrary real e,t, since the real exponential is positive. For e=E_0 and t≥0 it is also at most one. Therefore T_E_0(t)>0, Spec T_E_0(t)⊆(0,1]. The two scalar multiples of H-eI commute, so the exponential defined by the power series obeys T_e(t+s)=T_e(t)T_e(s), T_e(0)=I. The semigroup identity is proved for all real t,s. Nonnegative time is needed for the upper spectral bound, not for the c…
Exact mathematical passage
\boxed{T_{E_0}(t)>0,\qquad \Spec T_{E_0}(t)\subseteq(0,1].}
T_{e_j}Use in source · 1 passage
Search aliases: T_e_j
Use in source · Read passage · 03_supplement_02.tex, line 17
0 · x≠ a. The seed is ψ_0=ket0. Define a translation operator by (T_a v)(x)=v(x-a). At position j, the local Pauli X is X_j=T_e_j, where e_j has a single nonzero binary coordinate. The formal circuit uses the identity when a_j=0 and X_j when a_j=1.
Exact mathematical passage
X_j=T_{e_j}
t_jUse in source · 5 passages
Search aliases: t_j
Use in source · monograph_atlas.tex, line 687
4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phase…
Exact mathematical passage
t_j
t_{j+1}Use in source · 1 passage
Search aliases: t_j+1
Use in source · Read passage · 02_strengthened.tex, line 837
es S( P(t+h), P(t)) = I+hK(t)+O(h^2) uniformly on [0,T]. Since K(t)^*=-K(t), S^*S=I+O(h^2), so the polar factor satisfies U( P(t+h), P(t)) = I+hK(t)+O(h^2). This gives a first order approximation to dot W=KW over a single time step. A telescoping product estimate over the partition gives a total error bounded by a constant times ∑_j(t_j+1-t_j)^2, which is at most Tabsπ.
Exact mathematical passage
\sum_j(t_{j+1}-t_j)^2
t_kUse in source · 4 passages
Search aliases: t_k
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
t_mUse in source · 1 passage
Search aliases: t_m
Use in source · Read passage · 02_strengthened.tex, line 1713
[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.
Exact mathematical passage
t_1,\ldots,t_m
T_nUse in source · 1 passage
Search aliases: T_n
Use in source · monograph_atlas.tex, line 797
For transfers on two states T_n=diag(1,q_n)=e^-H_n, with H_n=diag(0,-log q_n) at unit time, the error in trace norm is 2^-(n+2) and the heat traces converge to 3/2. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat trace alone does not determine that quotient. headThe retained evidence and the remaining open targets The extension includes retained exact checks ove…
Exact mathematical passage
T_n=\diag(1,q_n)=e^{-H_n}
t_NUse in source · 1 passage
Search aliases: t_N
Use in source · monograph_atlas.tex, line 740
It does not assert a uniform interval width from that argument, a moving time t_N with no extra estimates, or a dimension plateau at every scale. At fixed finite N, d_N(t)→0 as tdownarrow0 and as t→∞. By contrast, first taking N→∞ at positive t and then taking tdownarrow0 yields the continuum dimension. headWhere the infinite tail is controlled At a fixed time, the proof bounds centered modes below by 16k^2 and above by 4π^2k^2. The summable Gaussian dominators control both the heat trace and it…
Exact mathematical passage
t_N
t_nUse in source · 2 passages
Search aliases: t_n
Use in source · Read passage · 02_strengthened.tex, line 800
[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
0=t_0<t_1<\cdots<t_n=T,
t_{n-1}Use in source · 1 passage
Search aliases: t_n-1
Use in source · Read passage · 02_strengthened.tex, line 804
[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
W_\pi = \Ucal\bigl(\Pcal(t_n),\Pcal(t_{n-1})\bigr) \cdots \Ucal\bigl(\Pcal(t_1),\Pcal(t_0)\bigr).
T_PUse in source · 3 passages
Search aliases: T_P
Use in source · Read passage · 03_supplement_04.tex, line 10
Deriving the estimate by differentiation. For a skew-adjoint generator G, define the actual conjugation path T_P(t)=exp(tG)Pexp(-tG). The formalization derives its derivative from the exponential power series and the product rule: T_P'(t)=exp(tG)(GP-PG)exp(-tG). Both exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore T_P'(t) = GP-PG , T_P(t)-T_P(s) ≤ |t-s| GP-PG . The second inequality follows from the mean value inequality in n…
Exact mathematical passage
T_P(t)=\exp(tG)P\exp(-tG).
T_P'Use in source · 2 passages
Search aliases: T_P'
Use in source · Read passage · 03_supplement_04.tex, line 14
Deriving the estimate by differentiation. For a skew-adjoint generator G, define the actual conjugation path T_P(t)=exp(tG)Pexp(-tG). The formalization derives its derivative from the exponential power series and the product rule: T_P'(t)=exp(tG)(GP-PG)exp(-tG). Both exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore T_P'(t) = GP-PG , T_P(t)-T_P(s) ≤ |t-s| GP-PG . The second inequality follows from the mean value inequality in n…
Exact mathematical passage
T_P'(t)=\exp(tG)(GP-PG)\exp(-tG).
t_wUse in source · 2 passages
Search aliases: t_w
Use in source · Read passage · 01_finite_core.tex, line 802
We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix A_w.
Exact mathematical passage
t_w\ge0
t_\varepsilonUse in source · 4 passages
Search aliases: t_ε varepsilon
Use in source · monograph_atlas.tex, line 738
4 · (0,32,32,64) · 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less than ε/2. Then choose N_ε so that every N≥ N_ε differs from that continuum value by less than ε/2 at the same time. This proves one controlled intermediate time f…
Exact mathematical passage
t_\varepsilon
\operatorname{Tr}Use in source · 1 passage
Search aliases: Tr operatorname
Use in source · trace_topology.tex, line 5
A_n _1=TrA_n=1; draw[->] (4.7,1.9)--(6.7,1.9);node[font=small] at (6,0.45) positive trace-class control: norm convergence alone does not preserve a trace;
Exact mathematical passage
\|A_n\|_1=\operatorname{Tr}A_n=1
\trUse in source · 5 passages
Search aliases: tr operatorname
Use in source · Read passage · 03_supplement_01.tex, line 79
Measurements: source Theorem 7.1. For finite complex matrices, define packet dephasing by D_P(X)=∑_aP_aXP_a. The code proves idempotence and trace duality: D_P( D_P(X))= D_P(X), tr( D_P(E)ρ)=tr(E D_P(ρ)). Trace duality follows term by term from cyclicity: tr(P_aEP_aρ)=tr(EP_aρ P_a). Therefore an effect that is diagonal in the packets, D_P(E)=E, satisfies tr(Eρ)=tr(E D_P(ρ)). The proof holds for arbitrary finite complex matrices E and ρ. Density matrices and positive measurement effects are valid…
Exact mathematical passage
\deph(\deph(X))=\deph(X),\qquad \tr\bigl(\deph(E)\rho\bigr)=\tr\bigl(E\deph(\rho)\bigr).
\Tr_AUse in source · 1 passage
Search aliases: Tr_A Tr operatorname
Use in source · Read passage · 01_finite_core.tex, line 137
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\rho_B=\Tr_A(\ket\psi\bra\psi)
\Tr_{\rm out}Use in source · 1 passage
Search aliases: Tr_rm out Tr rm operatorname rm
Use in source · Read passage · 02_strengthened.tex, line 1083
labelstrong:standardfact:001 A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.
Exact mathematical passage
\Tr_{\rm out}J(\Phi)=I_{\rm in}.
uUse in source · 27 passages
Search aliases: u
Use in source · monograph_atlas.tex, line 194
ORS AND PROBABILITY BOUNDSTraditional edition: Theorem II.5.3, Corollary II.5.4. For a two-level system, choose H=ε X, P=ket0bra0, and Q=I-P. Direct exponentiation gives e^- i tH=cos(ε t)I- isin(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently P…
Exact mathematical passage
|\sin u|\le |u|
U^*Use in source · 23 passages
Search aliases: U^*
Use in source · monograph_atlas.tex, line 217
1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i
Exact mathematical passage
U^*\dot U=J
U_1Use in source · 7 passages
Search aliases: U_1
Use in source · Read passage · 01_finite_core.tex, line 276
Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.
Exact mathematical passage
U=U_mU_{m-1}\cdots U_1
u_aUse in source · 3 passages
Search aliases: u_a
Use in source · monograph_atlas.tex, line 430
splayed T happens also to have nonnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivi…
Exact mathematical passage
u_a
U_{ar}Use in source · 2 passages
Search aliases: U_ar
Use in source · Read passage · 01_finite_core.tex, line 843
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
F_r=\sum_aU_{ar}E_aP.
U_bUse in source · 1 passage
Search aliases: U_b
Use in source · Read passage · 03_supplement_06.tex, line 49
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
U_a\cap U_b
u_bUse in source · 1 passage
Search aliases: u_b
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
U_{ba}Use in source · 7 passages
Search aliases: U_ba
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
U_{ba}^*Use in source · 3 passages
Search aliases: U_ba^*
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
U_{bc}Use in source · 5 passages
Search aliases: U_bc
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
U_{da}Use in source · 1 passage
Search aliases: U_da
Use in source · Read passage · 03_supplement_01.tex, line 35
Adjoint conservation: source Theorem 4.2. Assume a star operation with P_a^*=P_a. Then (U_ba)^*=P_aU^*P_b. The two checked conservation laws are U^*U=1 · Longrightarrow ∑_b(U_ba)^*U_bc=δ_acP_a, UU^*=1 · Longrightarrow ∑_aU_ba(U_da)^*=δ_bdP_b. For the first law, expand each corner, collapse P_b^2 and sum: ∑_bP_aU^*P_bUP_c =P_aU^*(∑_bP_b)UP_c=P_aP_c. The second law follows by the analogous calculation and has its own formal proof. For complex matrices, the adjoint is the conjugate transpose. The o…
Exact mathematical passage
U^*U=1&\quad\Longrightarrow\quad \sum_b(U_{ba})^*U_{bc}=\delta_{ac}P_a,\\ UU^*=1&\quad\Longrightarrow\quad \sum_aU_{ba}(U_{da})^*=\delta_{bd}P_b.
U_{da}^*Use in source · 1 passage
Search aliases: U_da^*
Use in source · Read passage · 02_strengthened.tex, line 424
[Packet unitarity identities] If U is unitary and U_ba=P_bUP_a, then ∑_bU_ba^*U_bc=δ_acP_a and ∑_aU_baU_da^*=δ_bdP_b.
Exact mathematical passage
\sum_aU_{ba}U_{da}^*=\delta_{bd}P_b.
u_jUse in source · 2 passages
Search aliases: u_j
Use in source · monograph_atlas.tex, line 716
trivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This …
Exact mathematical passage
u_j
u_kUse in source · 1 passage
Search aliases: u_k
Use in source · monograph_atlas.tex, line 717
tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This is a finite Gram factorization. The retained run checks the integer sem…
Exact mathematical passage
\sum_{j,k}\langle u_j,T^{n_j+n_k}u_k\rangle =\left\|\sum_jT^{n_j}u_j\right\|^2.
U_mUse in source · 7 passages
Search aliases: U_m
Use in source · Read passage · 01_finite_core.tex, line 276
Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.
Exact mathematical passage
U=U_mU_{m-1}\cdots U_1
U_{m-1}Use in source · 2 passages
Search aliases: U_m-1
Use in source · Read passage · 01_finite_core.tex, line 276
Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.
Exact mathematical passage
U=U_mU_{m-1}\cdots U_1
u_NUse in source · 3 passages
Search aliases: u_N
Use in source · monograph_atlas.tex, line 283
n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ…
Exact mathematical passage
u_N=(\cos(\theta/2),e^{\ii\phi}\sin(\theta/2))^T
u_N^*Use in source · 2 passages
Search aliases: u_N^*
Use in source · monograph_atlas.tex, line 283
ojector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ≤θ_0 is π(1-cosθ_0). At θ_0=π/3, it is exactl…
Exact mathematical passage
P=u_Nu_N^*
U_rUse in source · 1 passage
Search aliases: U_r
Use in source · Read passage · 02_strengthened.tex, line 485
[Charge pruning of circuit paths] Suppose every gate has a finite charge decomposition U_r=∑_qU_r^(q), SU_r^(q)S^*=ω^qU_r^(q). A term with gate charges q_1,…,q_m can contribute from source charge a_0 only to target charge a_m=a_0+q_1+⋯+q_m. Every packet path violating one of the intermediate charge updates vanishes.
Exact mathematical passage
U_r=\sum_qU_r^{(q)}, \qquad SU_r^{(q)}S^*=\omega^qU_r^{(q)}.
U_r^{(q)}Use in source · 1 passage
Search aliases: U_r^(q)
Use in source · Read passage · 02_strengthened.tex, line 485
[Charge pruning of circuit paths] Suppose every gate has a finite charge decomposition U_r=∑_qU_r^(q), SU_r^(q)S^*=ω^qU_r^(q). A term with gate charges q_1,…,q_m can contribute from source charge a_0 only to target charge a_m=a_0+q_1+⋯+q_m. Every packet path violating one of the intermediate charge updates vanishes.
Exact mathematical passage
U_r=\sum_qU_r^{(q)}, \qquad SU_r^{(q)}S^*=\omega^qU_r^{(q)}.
U_{\beta\alpha}Use in source · 2 passages
Search aliases: U_βα beta alpha
Use in source · Read passage · 01_finite_core.tex, line 301
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
U_{\beta\alpha}=P_\beta UP_\alpha
U_{\beta\alpha}^*Use in source · 2 passages
Search aliases: U_βα^* beta alpha
Use in source · Read passage · 01_finite_core.tex, line 302
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\boxed{ \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma} = \delta_{\alpha\gamma}P_\alpha. }
U_{\beta\gamma}Use in source · 2 passages
Search aliases: U_βγ beta gamma
Use in source · Read passage · 01_finite_core.tex, line 302
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\boxed{ \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma} = \delta_{\alpha\gamma}P_\alpha. }
U_\gammaUse in source · 2 passages
Search aliases: U_γ gamma
Use in source · Read passage · 02_strengthened.tex, line 410
For a packet path γ=(a_0,a_1,…,a_m), write U_γ = P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0.
Exact mathematical passage
U_\gamma = P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
U_{\delta\alpha}^*Use in source · 1 passage
Search aliases: U_δα^* delta alpha
Use in source · Read passage · 01_finite_core.tex, line 310
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\sum_\alpha U_{\beta\alpha}U_{\delta\alpha}^* = \delta_{\beta\delta}P_\beta.
\mathrm{unused}Use in source · 2 passages
Search aliases: unused mathrm
Use in source · Read passage · 01_finite_core.tex, line 851
efine compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP…
Exact mathematical passage
K_r=V_r^*,\qquad K_{\mathrm{unused}}=I-S.
V'Use in source · 1 passage
Search aliases: V'
Use in source · Read passage · 02_strengthened.tex, line 862
[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.
Exact mathematical passage
V'=VG
V^*Use in source · 8 passages
Search aliases: V^*
Use in source · monograph_atlas.tex, line 370
main inside the original code, so the previous syndrome partition reports no bit flip. frac3ket000+4ket1115 ⟼ frac3ket000-4ket1115. The populations are unchanged, but the relative phase is not. vfiglogical_error Let V be the code isometry with columns ket000,ket111. For the proposed error family {I,Z_1}, its off-diagonal error compression is V^*I^*Z_1V=mat1 · 0 0 · -1. This matrix is not cI for any scalar c. Thus the Knill--Laflamme condition fails for that family. Its trace is zero even though …
Exact mathematical passage
V^*I^*Z_1V=\mat{1&0\\0&-1}.
v^*Use in source · 1 passage
Search aliases: v^*
Use in source · euclidean.tex, line 22
spectrum {1,1/2}.; node[font=small,align=center,font=small] at (7,0.4) Positive-definite operator: v^*Tv>0 for v≠0. Not an entrywise test.;
Exact mathematical passage
v^*Tv>0
v_0Use in source · 2 passages
Search aliases: v_0
Use in source · cyclic_four.tex, line 44
anchor=east,font=scriptsize] at (0.85,1.2) site 3; node[font=small,align=center,font=scriptsize] at (1.4,4.23) j=0; node[font=small,align=center,font=scriptsize] at (2.2,4.23) j=1; node[font=small,align=center,font=scriptsize] at (3,4.23) j=2; node[font=small,align=center,font=scriptsize] at (3.8,4.23) j=3; node[font=small,align=center,font=small] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95…
Exact mathematical passage
S v_0=1v_0
v_0^*Use in source · 1 passage
Search aliases: v_0^*
Use in source · cyclic_four.tex, line 45
gn=center,font=scriptsize] at (1.4,4.23) j=0; node[font=small,align=center,font=scriptsize] at (2.2,4.23) j=1; node[font=small,align=center,font=scriptsize] at (3,4.23) j=2; node[font=small,align=center,font=scriptsize] at (3.8,4.23) j=3; node[font=small,align=center,font=small] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95)…
Exact mathematical passage
P_0=v_0v_0^*
V_1Use in source · 1 passage
Search aliases: V_1
Use in source · monograph_atlas.tex, line 610
0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divide by zero.
Exact mathematical passage
V_1=P
v_1Use in source · 4 passages
Search aliases: v_1
Use in source · Read passage · 02_strengthened.tex, line 1713
[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.
Exact mathematical passage
v_1,\ldots,v_m
v_1^*Use in source · 1 passage
Search aliases: v_1^*
Use in source · cyclic_four.tex, line 47
,font=scriptsize] at (3,4.23) j=2; node[font=small,align=center,font=scriptsize] at (3.8,4.23) j=3; node[font=small,align=center,font=small] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) …
Exact mathematical passage
P_1=v_1v_1^*
v_2Use in source · 2 passages
Search aliases: v_2
Use in source · cyclic_four.tex, line 48
scriptsize] at (3.8,4.23) j=3; node[font=small,align=center,font=small] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at (8.05,1.25)…
Exact mathematical passage
S v_2=-1v_2
v_2^*Use in source · 1 passage
Search aliases: v_2^*
Use in source · cyclic_four.tex, line 49
ll] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at (8.05,1.25) S v_3=- iv_3; node[font=small,align=center,font=small] at (11.85,1.…
Exact mathematical passage
P_2=v_2v_2^*
v_3Use in source · 2 passages
Search aliases: v_3
Use in source · cyclic_four.tex, line 50
at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at (8.05,1.25) S v_3=- iv_3; node[font=small,align=center,font=small] at (11.85,1.25) P_3=v_3v_3^*; node[font=small,align=center,font=scriptsize] at (8.8…
Exact mathematical passage
S v_3=- iv_3
v_3^*Use in source · 1 passage
Search aliases: v_3^*
Use in source · cyclic_four.tex, line 51
at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at (8.05,1.25) S v_3=- iv_3; node[font=small,align=center,font=small] at (11.85,1.25) P_3=v_3v_3^*; node[font=small,align=center,font=scriptsize] at (8.8,0.12) Each column has four entries of magnitude 1/2.;
Exact mathematical passage
P_3=v_3v_3^*
V_aUse in source · 9 passages
Search aliases: V_a
Use in source · Read passage · 03_supplement_04.tex, line 94
A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.
Exact mathematical passage
V_a
v_aUse in source · 1 passage
Search aliases: v_a
Use in source · Read passage · 03_supplement_03.tex, line 53
An independent definition of Schmidt rank. The source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length s is the concrete equality C_ij=∑_a=1^su_a(i)v_a(j). It factors C=UV through an s-dimensional intermediate space, so rank C≤ s. Conversely, choose a basis of the row space of C. Expanding each row in this basis constructs a decomposition with exactly rank C product terms. Thus r(ψ)=rank C. The definition uses a least natural number, and th…
Exact mathematical passage
C_{ij}=\sum_{a=1}^{s}u_a(i)v_a(j).
V_a^*Use in source · 7 passages
Search aliases: V_a^*
Use in source · Read passage · 03_supplement_04.tex, line 95
A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.
Exact mathematical passage
V_aP=V_a, \qquad V_a^*V_b=\delta_{ab}P.
V_bUse in source · 1 passage
Search aliases: V_b
Use in source · Read passage · 03_supplement_04.tex, line 95
A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.
Exact mathematical passage
V_aP=V_a, \qquad V_a^*V_b=\delta_{ab}P.
V_b^*Use in source · 1 passage
Search aliases: V_b^*
Use in source · Read passage · 03_supplement_04.tex, line 122
Exact correction, including coherent errors. For every operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent sup…
Exact mathematical passage
\boxed{\mathcal R(V_a\rho V_b^*)=\delta_{ab}\rho.}
V_{ca}Use in source · 1 passage
Search aliases: V_ca
Use in source · Read passage · 03_supplement_01.tex, line 12
0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.
Exact mathematical passage
\sum_{b,a}U_{ba}=U,\qquad (UV)_{ba}=\sum_c U_{bc}V_{ca}.
V_jUse in source · 8 passages
Search aliases: V_j
Use in source · Read passage · 02_strengthened.tex, line 1327
syndrome construction] Assume the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
V_j=\lambda_j^{-1/2}F_jP
v_jUse in source · 10 passages
Search aliases: v_j
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
V_j^*Use in source · 6 passages
Search aliases: V_j^*
Use in source · Read passage · 02_strengthened.tex, line 1331
the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
Q_j=V_jV_j^*
v_j^*Use in source · 1 passage
Search aliases: v_j^*
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
v_kUse in source · 9 passages
Search aliases: v_k
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
v_mUse in source · 2 passages
Search aliases: v_m
Use in source · Read passage · 02_strengthened.tex, line 1713
[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.
Exact mathematical passage
v_1,\ldots,v_m
V_nUse in source · 2 passages
Search aliases: V_n
Use in source · Read passage · 03_supplement_02.tex, line 5
For n≥ 0, let A_n=( Z/2 Z)^{0,…,n-1}, V_n={v:A_n→ C}. For an address a∈ A_n, define the standard coordinate vector ket a(x)=1 · x=a,
Exact mathematical passage
A_n=(\mathbb Z/2\mathbb Z)^{\{0,\ldots,n-1\}},\qquad V_n=\{v:A_n\to\C\}.
V_sUse in source · 2 passages
Search aliases: V_s
Use in source · Read passage · 01_finite_core.tex, line 847
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
V_rP=V_r,\qquad V_r^*V_s=\delta_{rs}P.
V_\muUse in source · 1 passage
Search aliases: V_μ mu
Use in source · Read passage · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
V_\ellUse in source · 1 passage
Search aliases: V_ℓ ell
Use in source · Read passage · 02_strengthened.tex, line 1349
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
V_j^*V_\ell=\delta_{j\ell}P.
\operatorname{val}Use in source · 4 passages
Search aliases: val operatorname
Use in source · Read passage · 03_supplement_03.tex, line 93
Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N is proved from S^N=I, giving ρ(r+s)=ρ(r)ρ(s) and ρ(0)=I. The final matrix theorem derives the representation law …
Exact mathematical passage
\omega=\exp(2\pi i/N),\qquad \chi(r)=\omega^{\operatorname{val}(r)}.
WUse in source · 32 passages
Search aliases: W
Use in source · monograph_atlas.tex, line 45
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'…
Exact mathematical passage
W_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}. \qquad W_a\psi_0=\ket{a_1a_2a_3}.
W^*Use in source · 6 passages
Search aliases: W^*
Use in source · monograph_atlas.tex, line 603
0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divid…
Exact mathematical passage
W=\mat{3/5&4/5&0\\-4/5&3/5&0\\0&0&1}, \quad WcW^*=\diag(1,0,0), \quad(F_1,F_2,F_3)=(P,0,0).
w^*Use in source · 3 passages
Search aliases: w^*
Use in source · monograph_atlas.tex, line 338
0 · 0 · 0 · 1. Its middle block has one negative eigenvalue. More concretely, the normalized vector w=(ket01-ket10)/sqrt2 satisfies w^*(id⊗ T)(ρ_rm Bell)w=-frac12. This exact negative value of the quadratic form certifies failure of positivity on the composite system. No numerical eigenvalue tolerance is needed. headWhy the Kraus form avoids this problem For every auxiliary dimension and positive X, each term (I⊗ K_a)X(I⊗ K_a)^* is positive. Their sum is positive as well. This argument explains …
Exact mathematical passage
w^*(\id\otimes\mathsf T)(\rho_{\rm Bell})w=-\frac12.
w^2Use in source · 1 passage
Search aliases: w^2
Use in source · Read passage · 02_strengthened.tex, line 1727
Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.
Exact mathematical passage
\norm w^2 = \sum_{j,k} \ip{B^*v_j}{\e^{-\ii(t_j-t_k)D}B^*v_k} = \sum_{j,k} \ip{v_j}{K(t_j-t_k)v_k}.
W_{011}Use in source · 1 passage
Search aliases: W_011
Use in source · addresses.tex, line 33
node[font=small,align=center] at (2.65,4.05) ket110; path[draw,fill=white] (5.53,3.75) rectangle (6.47,4.35); node[font=small,align=center] at (6,4.05) ket111; node[font=small,align=center,font=small] at (10.15,4.1) Three actual words; node[font=small,align=center,font=small] at (10.15,3.1) W_101=X_1X_3; node[font=small,align=center,font=small] at (10.15,2.2) W_011=X_2X_3; node[font=small,align=center,font=small] at (10.15,1.3) W_110=X_1X_2; node[font=small,align=center,font=small] at (6.65,0.1)…
Exact mathematical passage
W_{011}=X_2X_3
W_{101}Use in source · 1 passage
Search aliases: W_101
Use in source · addresses.tex, line 32
,1.6) ket101; path[draw,fill=white] (2.18,3.75) rectangle (3.12,4.35); node[font=small,align=center] at (2.65,4.05) ket110; path[draw,fill=white] (5.53,3.75) rectangle (6.47,4.35); node[font=small,align=center] at (6,4.05) ket111; node[font=small,align=center,font=small] at (10.15,4.1) Three actual words; node[font=small,align=center,font=small] at (10.15,3.1) W_101=X_1X_3; node[font=small,align=center,font=small] at (10.15,2.2) W_011=X_2X_3; node[font=small,align=center,font=small] at (10.15,1.…
Exact mathematical passage
W_{101}=X_1X_3
W_{110}Use in source · 1 passage
Search aliases: W_110
Use in source · addresses.tex, line 34
hite] (5.53,3.75) rectangle (6.47,4.35); node[font=small,align=center] at (6,4.05) ket111; node[font=small,align=center,font=small] at (10.15,4.1) Three actual words; node[font=small,align=center,font=small] at (10.15,3.1) W_101=X_1X_3; node[font=small,align=center,font=small] at (10.15,2.2) W_011=X_2X_3; node[font=small,align=center,font=small] at (10.15,1.3) W_110=X_1X_2; node[font=small,align=center,font=small] at (6.65,0.1) dimSpan{W_aket000}=8;
Exact mathematical passage
W_{110}=X_1X_2
W_aUse in source · 6 passages
Search aliases: W_a
Use in source · monograph_atlas.tex, line 45
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'…
Exact mathematical passage
W_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}. \qquad W_a\psi_0=\ket{a_1a_2a_3}.
w_aUse in source · 3 passages
Search aliases: w_a
Use in source · monograph_atlas.tex, line 351
II.8.4, Example II.8.8. Encode a qubit by ket0_L=ket000 and ket1_L=ket111. Let P project onto this code and take errors E_a∈{I,X_1,X_2,X_3}. Their images are four orthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an ac…
Exact mathematical passage
\sum_aw_aE_a\psi\psi^*E_a^*
W_\piUse in source · 2 passages
Search aliases: W_π pi
Use in source · Read passage · 02_strengthened.tex, line 804
[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
W_\pi = \Ucal\bigl(\Pcal(t_n),\Pcal(t_{n-1})\bigr) \cdots \Ucal\bigl(\Pcal(t_1),\Pcal(t_0)\bigr).
\widehat CUse in source · 3 passages
Search aliases: widehat C widehat
Use in source · Read passage · 01_finite_core.tex, line 1198
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\widehat C_A
\widehat C_AUse in source · 2 passages
Search aliases: widehat C_A widehat
Use in source · Read passage · 01_finite_core.tex, line 1198
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\widehat C_A
\widehat C_BUse in source · 1 passage
Search aliases: widehat C_B widehat
Use in source · Read passage · 01_finite_core.tex, line 1199
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\boxed{ \mathfrak A_{AB} = [\widehat C_A,\widehat C_B] -i\sum_C f_{AB}^{\ \ C}\widehat C_C. }
\widehat C_CUse in source · 2 passages
Search aliases: widehat C_C widehat
Use in source · Read passage · 01_finite_core.tex, line 1199
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\boxed{ \mathfrak A_{AB} = [\widehat C_A,\widehat C_B] -i\sum_C f_{AB}^{\ \ C}\widehat C_C. }
\widetilde BUse in source · 9 passages
Search aliases: widetilde B widetilde
Use in source · monograph_atlas.tex, line 660
In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. Then |Σ-widetildeΣ|=|1-frac81…
Exact mathematical passage
(\widetilde B,\widetilde D)=(9/8,17/8)
\widetilde B^*Use in source · 2 passages
Search aliases: widetilde B^* widetilde
Use in source · Read passage · 02_strengthened.tex, line 1787
[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D) ≥δ>0. Then normΣ(z)-widetildeΣ(z) ≤ frac2gδnormB-widetilde B + fracg^2δ^2normD-widetilde D.
Exact mathematical passage
\Sigma(z)=B(z-D)^{-1}B^*, \qquad \widetilde\Sigma(z)=\widetilde B(z-\widetilde D)^{-1}\widetilde B^*.
\widetilde CUse in source · 5 passages
Search aliases: widetilde C widetilde
Use in source · monograph_atlas.tex, line 753
ull algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into th…
Exact mathematical passage
\widetilde C_a=C_a\oplus1
\widetilde C_aUse in source · 3 passages
Search aliases: widetilde C_a widetilde
Use in source · monograph_atlas.tex, line 753
ull algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into th…
Exact mathematical passage
\widetilde C_a=C_a\oplus1
\widetilde C_bUse in source · 1 passage
Search aliases: widetilde C_b widetilde
Use in source · monograph_atlas.tex, line 754
C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into the fine common kernel. However, the same prescribed Pa…
Exact mathematical passage
[\widetilde C_a,\widetilde C_b]R=R[C_a,C_b].
\widetilde C_xUse in source · 1 passage
Search aliases: widetilde C_x widetilde
Use in source · monograph_atlas.tex, line 759
e coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does not close on the whole fine level: on the added coordinate the commutator is zero, whereas 2 iwidetilde C_z is 2 i. Exact transport constrains the image of R and leaves its orthogonal complement as separate data. headAn approximate residual is also explicit Perturb only widetilde C_x by (1/10)ket0bra0. For the unit coarse physical vector, the fine constraint residual is exactly 1/10. The …
Exact mathematical passage
\widetilde C_x
\widetilde C_zUse in source · 1 passage
Search aliases: widetilde C_z widetilde
Use in source · monograph_atlas.tex, line 757
the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does not close on the whole fine level: on the added coordinate the commutator is zero, whereas 2 iwidetilde C_z is 2 i. Exact transport constrains the image of R and leaves its orthogonal complement as separate data. headAn approx…
Exact mathematical passage
2\ii\widetilde C_z
\widetilde DUse in source · 8 passages
Search aliases: widetilde D widetilde
Use in source · monograph_atlas.tex, line 660
In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. Then |Σ-widetildeΣ|=|1-frac81…
Exact mathematical passage
(\widetilde B,\widetilde D)=(9/8,17/8)
\widetilde D^*Use in source · 1 passage
Search aliases: widetilde D^* widetilde
Use in source · Read passage · 02_strengthened.tex, line 1786
[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D) ≥δ>0. Then normΣ(z)-widetildeΣ(z) ≤ frac2gδnormB-widetilde B + fracg^2δ^2normD-widetilde D.
Exact mathematical passage
\widetilde D=\widetilde D^*
\widetilde\SigmaUse in source · 4 passages
Search aliases: widetildeΣ widetilde Sigma
Use in source · monograph_atlas.tex, line 661
The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. Then |Σ-widetildeΣ|=|1-frac8156|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the diff…
Exact mathematical passage
|\Sigma-\widetilde\Sigma|=\left|1-\frac{81}{56}\right|=\frac{25}{56}, \qquad \frac{2g}{\delta}|B-\widetilde B| +\frac{g^2}{\delta^2}|D-\widetilde D|=\frac{21}{32}.
X^*Use in source · 4 passages
Search aliases: X^*
Use in source · Read passage · 02_strengthened.tex, line 939
Density matrices and observables carry both an input packet and an output packet. Equip End( H) with the Hilbert Schmidt inner product ipXY_HS=Tr(X^*Y).
Exact mathematical passage
\ip{X}{Y}_{\HS}=\Tr(X^*Y).
x^*Use in source · 1 passage
Search aliases: x^*
Use in source · Read passage · 03_supplement_05.tex, line 135
Complex polarization reaches the whole operator space. Write q(x)=xx^*. The module proves the exact identity 4xy^*=q(x+y)-q(x-y) +i(q(x+iy)-q(x-iy)). Agreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by P proves equality on every code-supported operator, including non-Hermitian ones.
Exact mathematical passage
q(x)=xx^*
X^{-1}Use in source · 1 passage
Search aliases: X^-1
Use in source · Read passage · 02_strengthened.tex, line 1668
Use the inverse difference identity X^-1-Y^-1=X^-1(Y-X)Y^-1 with X=F_P(z) and Y=z-A.
Exact mathematical passage
X^{-1}-Y^{-1}=X^{-1}(Y-X)Y^{-1}
x^2Use in source · 7 passages
Search aliases: x^2
Use in source · Read passage · 01_finite_core.tex, line 1155
Put x=kh/2. The inequalities |sin x|≤|x| and |x-sin x|≤ |x|^3/6, which hold for every real argument, give 0≤ x^2-sin^2x =|x-sin x| |x+sin x| ≤frac|x|^36 2|x| =fracx^43. Multiplying by 4/h^2 proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps k fixed while h→0, so it does not assert a uniformly small relative error across all grid modes.
Exact mathematical passage
0\le x^2-\sin^2x =|x-\sin x|\,|x+\sin x| \le\frac{|x|^3}{6}\,2|x| =\frac{x^4}{3}.
x^3Use in source · 3 passages
Search aliases: x^3
Use in source · Read passage · 02_strengthened.tex, line 2049
For x∈[0,π/2], sin x≤ x and sin x≥ x-fracx^36. Thus 0≤ x^2-sin^2x≤fracx^43. Apply this with x=πabsk_j/N, multiply by 4N^2, and sum over j.
Exact mathematical passage
\sin x\geq x-\frac{x^3}{6}.
x^4Use in source · 7 passages
Search aliases: x^4
Use in source · Read passage · 01_finite_core.tex, line 1155
Put x=kh/2. The inequalities |sin x|≤|x| and |x-sin x|≤ |x|^3/6, which hold for every real argument, give 0≤ x^2-sin^2x =|x-sin x| |x+sin x| ≤frac|x|^36 2|x| =fracx^43. Multiplying by 4/h^2 proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps k fixed while h→0, so it does not assert a uniformly small relative error across all grid modes.
Exact mathematical passage
0\le x^2-\sin^2x =|x-\sin x|\,|x+\sin x| \le\frac{|x|^3}{6}\,2|x| =\frac{x^4}{3}.
X_{(i,\cdot),(j,\cdot)}Use in source · 1 passage
Search aliases: X_(i,·),(j,·) cdot cdot
Use in source · Read passage · 03_supplement_02.tex, line 206
To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive …
Exact mathematical passage
[\Phi^{(J)}(X)]_{(i,u),(j,v)} =\Phi\bigl(X_{(i,\cdot),(j,\cdot)}\bigr)_{u,v}.
X_1Use in source · 6 passages
Search aliases: X_1
Use in source · monograph_atlas.tex, line 38
bottomrule headNormalization and indexing The tensor factors are ordered from left to right, so X_1=X⊗ I⊗ I. A normalized pure state gives ρ=ψψ^*. A coefficient matrix used to count a seed orbit need not be normalized. Scaling a nonzero seed does not change its generated linear span.
Exact mathematical passage
X_1=X\otimes I\otimes I
X_2Use in source · 5 passages
Search aliases: X_2
Use in source · monograph_atlas.tex, line 44
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'…
Exact mathematical passage
X_1,X_2,X_3
X_2^{a_2}Use in source · 1 passage
Search aliases: X_2^a_2
Use in source · monograph_atlas.tex, line 45
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'…
Exact mathematical passage
W_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}. \qquad W_a\psi_0=\ket{a_1a_2a_3}.
X_3Use in source · 5 passages
Search aliases: X_3
Use in source · monograph_atlas.tex, line 44
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'…
Exact mathematical passage
X_1,X_2,X_3
X_3^{a_3}Use in source · 1 passage
Search aliases: X_3^a_3
Use in source · monograph_atlas.tex, line 45
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'…
Exact mathematical passage
W_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}. \qquad W_a\psi_0=\ket{a_1a_2a_3}.
x_iUse in source · 1 passage
Search aliases: x_i
Use in source · Read passage · 03_supplement_02.tex, line 48
Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite number of factors with the stated finite coordinate sets.
Exact mathematical passage
\mathsf T(A)(x,y)=\prod_{i\in F} A_i(x_i,y_i).
X_j^{a_j}Use in source · 1 passage
Search aliases: X_j^a_j
Use in source · Read passage · 03_supplement_02.tex, line 21
For every binary address a, W_a=∏_j=0^n-1X_j^a_j=T_a, W_aψ_0=ket a. The family (W_aψ_0)_a∈ A_n is linearly independent, spans V_n, and has exactly 2^n elements. It is the standard computational basis.
Exact mathematical passage
W_a=\prod_{j=0}^{n-1}X_j^{a_j}=T_a, \qquad W_a\psi_0=\ket a.
\xmapstoUse in source · 1 passage
Search aliases: xmapsto
Use in source · monograph_atlas.tex, line 126
vpage8Track the Bell state through the circuitINTERMEDIATE PACKETS ARE NOT FINAL PACKETSTraditional edition: Theorem II.4.1, Example II.4.6. The input is ket00. First apply a Hadamard to the first qubit, then a controlled flip: ket00 xmapstoH_rm Had⊗ I fracket00+ket10sqrt2 xmapstorm CNOT fracket00+ket11sqrt2. The displayed columns keep amplitudes, including their normalization, at every step. The middle vector occupies both parity packets. The final vector occupies only the even packet. vfigbell…
Exact mathematical passage
\ket{00}\ \xmapsto{H_{\rm Had}\otimes I}\ \frac{\ket{00}+\ket{10}}{\sqrt2} \ \xmapsto{\rm CNOT}\ \frac{\ket{00}+\ket{11}}{\sqrt2}.
yUse in source · 40 passages
Search aliases: y
Use in source · monograph_atlas.tex, line 210
vpage12A moving frame creates a connection termNOT ALL MOTION IS PHYSICAL EVOLUTIONTraditional edition: Theorem II.6.3. Set the laboratory Hamiltonian to zero and keep ψ(t)=ket0 fixed. Describe it in rotating coordinates, U(t)=matcos t · -sin t sin t · cos t, y(t)=U(t)^*ψ(t)=matcos t -sin t. The vector has not moved in the laboratory, but its coordinate entries have changed. vfigmoving_frame headThe missing term can be computed exactly Let J=mat0 · -1
Exact mathematical passage
U(t)=\mat{\cos t&-\sin t\\\sin t&\cos t}, \qquad y(t)=U(t)^*\psi(t)=\mat{\cos t\\-\sin t}.
y^*Use in source · 1 passage
Search aliases: y^*
Use in source · Read passage · 03_supplement_05.tex, line 136
Complex polarization reaches the whole operator space. Write q(x)=xx^*. The module proves the exact identity 4xy^*=q(x+y)-q(x-y) +i(q(x+iy)-q(x-iy)). Agreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by P proves equality on every code-supported operator, including non-Hermitian ones.
Exact mathematical passage
4xy^*=q(x+y)-q(x-y) +i\bigl(q(x+iy)-q(x-iy)\bigr).
Y^{-1}Use in source · 1 passage
Search aliases: Y^-1
Use in source · Read passage · 02_strengthened.tex, line 1668
Use the inverse difference identity X^-1-Y^-1=X^-1(Y-X)Y^-1 with X=F_P(z) and Y=z-A.
Exact mathematical passage
X^{-1}-Y^{-1}=X^{-1}(Y-X)Y^{-1}
y_iUse in source · 1 passage
Search aliases: y_i
Use in source · Read passage · 03_supplement_02.tex, line 48
Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite number of factors with the stated finite coordinate sets.
Exact mathematical passage
\mathsf T(A)(x,y)=\prod_{i\in F} A_i(x_i,y_i).
\mathbb ZUse in source · 5 passages
Search aliases: Z mathbb
Use in source · Read passage · 03_supplement_02.tex, line 5
For n≥ 0, let A_n=( Z/2 Z)^{0,…,n-1}, V_n={v:A_n→ C}. For an address a∈ A_n, define the standard coordinate vector ket a(x)=1 · x=a,
Exact mathematical passage
A_n=(\mathbb Z/2\mathbb Z)^{\{0,\ldots,n-1\}},\qquad V_n=\{v:A_n\to\C\}.
Z^*Use in source · 2 passages
Search aliases: Z^*
Use in source · Read passage · 03_supplement_05.tex, line 111
Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this calculation with the original error and recovery matrices.
Exact mathematical passage
PE_a^*E_bP &=PE_a^*\Bigl(\sum_rR_r^*R_r\Bigr)E_bP\\ &=\sum_r(R_rE_aP)^*(R_rE_bP)\\ &=\Bigl(\sum_r\overline{z_{ra}}z_{rb}\Bigr)P =(Z^*Z)_{ab}P.
z^*Use in source · 1 passage
Search aliases: z^*
Use in source · Read passage · 02_strengthened.tex, line 1339
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_z^*F_zP=(z^*cz)P\geq0,
z^2Use in source · 1 passage
Search aliases: z^2
Use in source · monograph_atlas.tex, line 382
1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. This is the corrected hypothesis set recorded in the formal release. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, an error of 1/6. The omitted coupling has changed the response even though the retained diagonal entry did not change. headA pole can bel…
Exact mathematical passage
H_{\rm eff}(z)=\frac1{z-2},\qquad \bigl[(zI-H)^{-1}\bigr]_{00} =\frac1{z-1/(z-2)}=\frac{z-2}{z^2-2z-1}.
z^nUse in source · 3 passages
Search aliases: z^n
Use in source · monograph_atlas.tex, line 670
|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings. headA series near zero energy and its exact remainder At z=1/2, |z| |D^-1|=1/4<1, so Σ(z)=-∑_n=0^∞fracz^n2^n+1. After the first four terms, the exact remainder has magnitude 1/384. The geometric series controls the infinite tail. The retained che…
Exact mathematical passage
\Sigma(z)=-\sum_{n=0}^{\infty}\frac{z^n}{2^{n+1}}.
Z^{\otimes n}Use in source · 1 passage
Search aliases: Z^⊗ n otimes
Use in source · Read passage · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
Z^{\otimes3}Use in source · 1 passage
Search aliases: Z^⊗3 otimes
Use in source · Read passage · 01_finite_core.tex, line 270
[Exact parity refinement] For three qubits, the local computational projectors were grouped by the parity of their Hamming weight. The resulting coarse projectors agreed with (I± Z^⊗3)/2 with residual 0.
Exact mathematical passage
(I\pm Z^{\otimes3})/2
Z_1Use in source · 6 passages
Search aliases: Z_1
Use in source · monograph_atlas.tex, line 361
vpage20An unchanged syndrome can hide a logical errorTHE SCALAR COMPRESSION TEST MATTERSTraditional edition: Theorem II.8.2, Proposition II.8.7. Apply Z_1=Z⊗ I⊗ I to the same repetition code. It sends ket000 to itself and ket111 to its negative. Both states remain inside the original code, so the previous syndrome partition reports no bit flip. frac3ket000+4ket1115 ⟼ frac3ket000-4ket1115. The populations are unchanged, but the relative phase is not. vfiglogical_error Let V be the code isometry w…
Exact mathematical passage
Z_1=Z\otimes I\otimes I
z_aUse in source · 3 passages
Search aliases: z_a
Use in source · Read passage · 02_strengthened.tex, line 1338
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
F_z=\sum_az_aE_a
Z_dUse in source · 3 passages
Search aliases: Z_d
Use in source · Read passage · 02_strengthened.tex, line 2107
[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.
Exact mathematical passage
Z_d(t) = \sum_{k\in\ZZ^d}\e^{-4\pi^2t\abs{k}^2} = (4\pi t)^{-d/2} \sum_{n\in\ZZ^d}\e^{-\abs{n}^2/(4t)}.
Z_{L,a}Use in source · 3 passages
Search aliases: Z_L,a
Use in source · Read passage · 02_strengthened.tex, line 1993
[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.
Exact mathematical passage
Z_L(t)=\sum_aZ_{L,a}(t), \qquad Z_{L,a}(t)=\Tr(P_a\e^{-tL}).
Z_NUse in source · 4 passages
Search aliases: Z_N
Use in source · monograph_atlas.tex, line 725
vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSTraditional edition: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).
Exact mathematical passage
Z_N(t)=\sum_ke^{-t\lambda_N(k)}
Z_{ra}Use in source · 1 passage
Search aliases: Z_ra
Use in source · Read passage · 03_supplement_05.tex, line 110
Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP
Exact mathematical passage
Z_{ra}=z_{ra}
z_{ra}Use in source · 3 passages
Search aliases: z_ra
Use in source · Read passage · 03_supplement_05.tex, line 103
Applied to the corrected Kraus family, this yields scalars z_ra with R_rE_aP=z_raP. The essential declarations are pathrank_one_gram_rigidity, pathkraus_rank_one_rigidity and pathcorrected_kraus_family. No rank-one Choi characterization is inserted as an axiom or as the desired conclusion in hypothesis form.
Exact mathematical passage
z_{ra}
z_{rb}Use in source · 1 passage
Search aliases: z_rb
Use in source · Read passage · 03_supplement_05.tex, line 111
Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this calculation with the original error and recovery matrices.
Exact mathematical passage
PE_a^*E_bP &=PE_a^*\Bigl(\sum_rR_r^*R_r\Bigr)E_bP\\ &=\sum_r(R_rE_aP)^*(R_rE_bP)\\ &=\Bigl(\sum_r\overline{z_{ra}}z_{rb}\Bigr)P =(Z^*Z)_{ab}P.
\ZZ^dUse in source · 4 passages
Search aliases: ZZ^d Z^d ZZ mathbb
Use in source · Read passage · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
|^2Use in source · 3 passages
Search aliases: |^2
Use in source · monograph_atlas.tex, line 156
ity. Starting from ket0, there are two resolved paths to output ket1. Their signed products are bra1H_rm Hadket0bra0H_rm Hadket0=frac12, bra1H_rm Hadket1bra1H_rm Hadket0=-frac12. Their sum is zero. Squaring each separately gives a different answer. vfiginterference headWhere the missing term goes For path amplitudes a,b, the outcome probability is |a+b|^2=|a|^2+|b|^2+2Re(abar b). Here the last term is -1/2, canceling the first two terms. The path picture retains this information only if its edge…
Exact mathematical passage
|a+b|^2=|a|^2+|b|^2+2\Re(a\bar b)
|^3Use in source · 3 passages
Search aliases: |^3
Use in source · Read passage · 01_finite_core.tex, line 1154
Put x=kh/2. The inequalities |sin x|≤|x| and |x-sin x|≤ |x|^3/6, which hold for every real argument, give 0≤ x^2-sin^2x =|x-sin x| |x+sin x| ≤frac|x|^36 2|x| =fracx^43. Multiplying by 4/h^2 proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps k fixed while h→0, so it does not assert a uniformly small relative error across all grid modes.
Exact mathematical passage
|x-\sin x|\le |x|^3/6
|_{U_a}Use in source · 1 passage
Search aliases: |_U_a
Use in source · Read passage · 03_supplement_06.tex, line 43
). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
Exact mathematical passage
\mathcal E|_{U_a}\ \cong\ U_a\times\mathcal E_a
\pmUse in source · 7 passages
Search aliases: ± pm
Use in source · monograph_atlas.tex, line 179
1 · 0. The even block squares to (25/16)I, so its eigenvalues are ±5/4. The odd block squares to I, so its eigenvalues are ±1. Exponentiating either block gives the corresponding part of the exact evolution. Transfer between the packets remains zero for every real time. headWhat has changed? Only the order of basis vectors has changed. No matrix entries, amplitudes, or energy levels were discarded. This is why the decomposition is exact. Keeping just the even block would be a separate restrictio…
Exact mathematical passage
\pm5/4
\cdotUse in source · 11 passages
Search aliases: · cdot
Use in source · monograph_atlas.tex, line 273
vpage15Curvature measures a local area responseAN EXACT BLOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away…
Exact mathematical passage
P(\theta,\phi)=\frac12(I+n\cdot\sigma).
\timesUse in source · 29 passages
Search aliases: × times
Use in source · monograph_atlas.tex, line 56
The exact check constructs each 8×8 word matrix, applies it to the seed, and verifies that the matrix of all eight outputs is the identity. It also tests the displayed superposition against every word output. No search over an unspecified gate set is involved. takeawayA cyclic seed generates a vector space under an algebra. That conclusion does not automatically provide a unitary preparation procedure for every vector in it.
Exact mathematical passage
8\times8
\alpha_0Use in source · 2 passages
Search aliases: α_0 alpha
Use in source · Read passage · 01_finite_core.tex, line 282
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\alpha_0
\alpha_1Use in source · 1 passage
Search aliases: α_1 alpha
Use in source · Read passage · 01_finite_core.tex, line 283
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\boxed{ P_{\alpha_m}UP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}U_mP_{\alpha_{m-1}} \cdots P_{\alpha_1}U_1P_{\alpha_0}. }
\alpha_mUse in source · 2 passages
Search aliases: α_m alpha
Use in source · Read passage · 01_finite_core.tex, line 282
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\alpha_m
\alpha_{m-1}Use in source · 1 passage
Search aliases: α_m-1 alpha
Use in source · Read passage · 01_finite_core.tex, line 283
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\boxed{ P_{\alpha_m}UP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}U_mP_{\alpha_{m-1}} \cdots P_{\alpha_1}U_1P_{\alpha_0}. }
\gammaUse in source · 19 passages
Search aliases: γ gamma
Use in source · monograph_atlas.tex, line 311
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0
Exact mathematical passage
\gamma=9/25
\gamma'Use in source · 1 passage
Search aliases: γ' gamma
Use in source · Read passage · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\Gamma_bUse in source · 4 passages
Search aliases: Γ_b Gamma
Use in source · Read passage · 02_strengthened.tex, line 452
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b
Exact mathematical passage
\Gamma_b
\delta^2Use in source · 4 passages
Search aliases: δ^2 delta
Use in source · monograph_atlas.tex, line 661
The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. Then |Σ-widetildeΣ|=|1-frac8156|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the diff…
Exact mathematical passage
|\Sigma-\widetilde\Sigma|=\left|1-\frac{81}{56}\right|=\frac{25}{56}, \qquad \frac{2g}{\delta}|B-\widetilde B| +\frac{g^2}{\delta^2}|D-\widetilde D|=\frac{21}{32}.
\delta_{ab}Use in source · 8 passages
Search aliases: δ_ab delta
Use in source · monograph_atlas.tex, line 355
The Knill--Laflamme compressions are PE_a^*E_bP=δ_abP. Their scalar form says that the error labels do not resolve the encoded vector. The exact checks evaluate all sixteen compressions, all sixteen syndrome products, the four probabilities, the recovery normalization, and the full recovered matrix.
Exact mathematical passage
PE_a^*E_bP=\delta_{ab}P
\delta_{ac}Use in source · 4 passages
Search aliases: δ_ac delta
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
\delta_{ba}Use in source · 1 passage
Search aliases: δ_ba delta
Use in source · Read passage · 02_strengthened.tex, line 386
Let U=U_mU_m-1⋯ U_1 be a finite circuit with m≥1, and let (P_a)_a∈ I be a complete orthogonal packet family. For the empty circuit, U=I and its transfer block is P_bP_a=δ_baP_a, whose endpoint sum reconstructs I.
Exact mathematical passage
P_bP_a=\delta_{ba}P_a
\delta_{bd}Use in source · 3 passages
Search aliases: δ_bd delta
Use in source · Read passage · 02_strengthened.tex, line 424
[Packet unitarity identities] If U is unitary and U_ba=P_bUP_a, then ∑_bU_ba^*U_bc=δ_acP_a and ∑_aU_baU_da^*=δ_bdP_b.
Exact mathematical passage
\sum_aU_{ba}U_{da}^*=\delta_{bd}P_b.
\delta_{jk}Use in source · 5 passages
Search aliases: δ_jk delta
Use in source · Read passage · 01_finite_core.tex, line 206
[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.
Exact mathematical passage
P_jP_k=\delta_{jk}P_j, \qquad \sum_{j=0}^{N-1}P_j=I, \qquad SP_j=\omega^jP_j.
\delta_{j\ell}Use in source · 2 passages
Search aliases: δ_jℓ delta ell
Use in source · Read passage · 02_strengthened.tex, line 1343
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_j^*F_\ell P = \lambda_j\delta_{j\ell}P.
\delta_{rs}Use in source · 5 passages
Search aliases: δ_rs delta
Use in source · Read passage · 01_finite_core.tex, line 846
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
F_r^*F_s=\lambda_r\delta_{rs}P
\delta_{\alpha\gamma}Use in source · 1 passage
Search aliases: δ_αγ delta alpha gamma
Use in source · Read passage · 01_finite_core.tex, line 302
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\boxed{ \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma} = \delta_{\alpha\gamma}P_\alpha. }
\delta_{\beta\alpha}Use in source · 1 passage
Search aliases: δ_βα delta beta alpha
Use in source · Read passage · 01_finite_core.tex, line 279
Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.
Exact mathematical passage
P_\beta P_\alpha=\delta_{\beta\alpha}P_\alpha
\delta_{\beta\delta}Use in source · 1 passage
Search aliases: δ_βδ delta beta delta
Use in source · Read passage · 01_finite_core.tex, line 310
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\sum_\alpha U_{\beta\alpha}U_{\delta\alpha}^* = \delta_{\beta\delta}P_\beta.
\delta_{\lambda\mu}Use in source · 2 passages
Search aliases: δ_λμ delta lambda mu
Use in source · Read passage · 02_strengthened.tex, line 215
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda, \qquad \sum_{\lambda\in\widehat G}P_\lambda=I_{\Hcal}.
\Delta_\ellUse in source · 2 passages
Search aliases: Δ_ℓ Delta ell
Use in source · Read passage · 01_finite_core.tex, line 1038
A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Exact mathematical passage
\Delta_\ell(D) =C_\ell D_{\ell+1}E_\ell-D_\ell.
\varepsilon^2Use in source · 2 passages
Search aliases: ε^2 varepsilon
Use in source · monograph_atlas.tex, line 198
in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε t=π/6, the amplitude is 1/2 and the probability is 1/4. Near zero, amplitudes are first order while escape probabilities are second order. Mixing thos…
Exact mathematical passage
\Pr(\text{hidden packet at }t)\le\min\{1,\varepsilon^2t^2\}.
\varepsilon_aUse in source · 3 passages
Search aliases: ε_a varepsilon
Use in source · Read passage · 02_strengthened.tex, line 2296
[Approximate constraint transport] Assume normC_ℓ+1,aR_ℓ-R_ℓ C_ℓ,a ≤ε_a and normC_ℓ,a,normC_ℓ+1,a≤ M. Then a coarse physical unit vector ψ satisfies normC_ℓ+1,aR_ℓψ≤ε_a. Also, · norm [C_ℓ+1,a,C_ℓ+1,b]R_ℓ -R_ℓ[C_ℓ,a,C_ℓ,b]
Exact mathematical passage
\norm{C_{\ell+1,a}R_\ell-R_\ell C_{\ell,a}} \leq\varepsilon_a
\varepsilon_bUse in source · 1 passage
Search aliases: ε_b varepsilon
Use in source · Read passage · 02_strengthened.tex, line 2309
[Approximate constraint transport] Assume normC_ℓ+1,aR_ℓ-R_ℓ C_ℓ,a ≤ε_a and normC_ℓ,a,normC_ℓ+1,a≤ M. Then a coarse physical unit vector ψ satisfies normC_ℓ+1,aR_ℓψ≤ε_a. Also, · norm [C_ℓ+1,a,C_ℓ+1,b]R_ℓ -R_ℓ[C_ℓ,a,C_ℓ,b] · ≤ 2M(ε_a+ε_b).
Exact mathematical passage
\begin{aligned} &\norm{ [C_{\ell+1,a},C_{\ell+1,b}]R_\ell -R_\ell[C_{\ell,a},C_{\ell,b}] }\\ &\hspace{4em}\leq 2M(\varepsilon_a+\varepsilon_b). \end{aligned}
\etaUse in source · 7 passages
Search aliases: η eta
Use in source · monograph_atlas.tex, line 503
actual amplitude vector is (- i,1, i,1)^T/2, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state. headA channel changes the conclusion predictably Before evolving, mix the seed density matrix with its complete momentum dephasing: ρ_η=(1-η)ket0bra0+η∑_jP_jket0bra0P_j =(1-η)ket0bra0+η I/4, quad0≤η≤1. At transfer time, the probability at th…
Exact mathematical passage
\rho_\eta=(1-\eta)\ket0\bra0+\eta\sum_jP_j\ket0\bra0P_j =(1-\eta)\ket0\bra0+\eta I/4, \quad0\le\eta\le1.
\eta_\gammaUse in source · 4 passages
Search aliases: η_γ eta gamma
Use in source · Read passage · 02_strengthened.tex, line 453
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b
Exact mathematical passage
\eta_\gamma=U_\gamma\psi.
\eta_{\gamma'}Use in source · 1 passage
Search aliases: η_γ' eta gamma
Use in source · Read passage · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\thetaUse in source · 27 passages
Search aliases: θ theta
Use in source · monograph_atlas.tex, line 272
vpage15Curvature measures a local area responseAN EXACT BLOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away…
Exact mathematical passage
n=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)
\theta_0Use in source · 3 passages
Search aliases: θ_0 theta
Use in source · monograph_atlas.tex, line 288
∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ≤θ_0 is π(1-cosθ_0). At θ_0=π/3, it is exactly π/2. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.
Exact mathematical passage
0\le\theta\le\theta_0
\iotaUse in source · 6 passages
Search aliases: ι iota
Use in source · Read passage · 02_strengthened.tex, line 2361
Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.
Exact mathematical passage
\iota_\ell:\Hcal_\ell\longrightarrow\Hcal_\infty
\iota_\ellUse in source · 6 passages
Search aliases: ι_ℓ iota ell
Use in source · Read passage · 02_strengthened.tex, line 2361
Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.
Exact mathematical passage
\iota_\ell:\Hcal_\ell\longrightarrow\Hcal_\infty
\kappaUse in source · 2 passages
Search aliases: κ kappa
Use in source · Read passage · 01_finite_core.tex, line 1079
The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
Exact mathematical passage
H_{nn}=2\kappa+V(n), \qquad H_{n,n+1}=H_{n+1,n}=-\kappa,
\lambda^{-1/2}Use in source · 1 passage
Search aliases: λ^-1/2 lambda
Use in source · monograph_atlas.tex, line 610
0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divide by zero.
Exact mathematical passage
\lambda^{-1/2}
\lambda_jUse in source · 14 passages
Search aliases: λ_j lambda
Use in source · Read passage · 01_finite_core.tex, line 1115
Let L≥0 be a finite Laplacian on N≥1 states, with eigenvalues λ_j and at least one zero eigenvalue. For diffusion time τ>0, define the normalized heat trace K(τ)=frac1N∑_j e^-τλ_j and the scale-dependent spectral dimension d_s(τ) =-2fracdlog K(τ)dlogτ. For a finite system, d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differentiation gives d_s(τ)=2τ frac∑_jλ_j e^-τλ_j∑_j e^-τλ_j. It tends to zero as τdownarrow0 and, under the stated assumption of a zero mode, as τ→∞. If a g…
Exact mathematical passage
\lambda_j
\lambda_j^{-1/2}Use in source · 1 passage
Search aliases: λ_j^-1/2 lambda
Use in source · Read passage · 02_strengthened.tex, line 1327
syndrome construction] Assume the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
V_j=\lambda_j^{-1/2}F_jP
\lambda_{k,h}Use in source · 9 passages
Search aliases: λ_k,h lambda
Use in source · monograph_atlas.tex, line 461
vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters because the displayed expression divides by h^2. vfigdispersion headWhat is certified in the displayed table? Take k=1 and h=1/4,1/8,1/16. For x=h/2∈[0,…
Exact mathematical passage
\lambda_{k,h}=\frac4{h^2}\sin^2\!\left(\frac{kh}{2}\right),\qquad h\ne0.
\lambda_{\min}Use in source · 1 passage
Search aliases: λ_min lambda min
Use in source · Read passage · 01_finite_core.tex, line 1131
d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differentiation gives d_s(τ)=2τ frac∑_jλ_j e^-τλ_j∑_j e^-τλ_j. It tends to zero as τdownarrow0 and, under the stated assumption of a zero mode, as τ→∞. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to 2τλ_min. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controll…
Exact mathematical passage
2\tau\lambda_{\min}
\Lambda_NUse in source · 1 passage
Search aliases: Λ_N Lambda
Use in source · Read passage · 02_strengthened.tex, line 2021
Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].
Exact mathematical passage
\Lambda_N=(\ZZ/N\ZZ)^d
\Lambda_N^*Use in source · 2 passages
Search aliases: Λ_N^* Lambda
Use in source · Read passage · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
\lambda_rUse in source · 12 passages
Search aliases: λ_r lambda
Use in source · Read passage · 01_finite_core.tex, line 842
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
C=U\operatorname{diag}(\lambda_r)U^*
\lambda_r^{-1/2}Use in source · 1 passage
Search aliases: λ_r^-1/2 lambda
Use in source · Read passage · 03_supplement_05.tex, line 61
Null directions are removed by proof. If λ_r=0, then B_r^*B_r=0. Positivity of the ordinary complex matrix inner product gives B_r=0. This is the checked declaration pathDiagonalErrors.zero_weight_error. The normalized syndrome operators are defined only on A_+={r:λ_r>0}, V_r=λ_r^-1/2B_r. Consequently no inverse square root at a null eigenvalue is needed. Lean proves V_rP=V_r, V_r^*V_s=δ_rsP. The previous block's recovery is now instantiated from these derived syndromes, rather than from an assu…
Exact mathematical passage
A_+=\{r:\lambda_r>0\},\qquad V_r=\lambda_r^{-1/2}B_r.
\muUse in source · 7 passages
Search aliases: μ mu
Use in source · Read passage · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\nuUse in source · 1 passage
Search aliases: ν nu
Use in source · Read passage · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\PiUse in source · 5 passages
Search aliases: Π Pi
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\pi^2Use in source · 9 passages
Search aliases: π^2 pi
Use in source · monograph_atlas.tex, line 471
Each interval lies strictly inside (0,h^2/12). Dividing successive interval endpoints proves that both error ratios are between 3.99 and 4. The check verifies these inequalities by rational arithmetic, without relying on approximate sine values or a fitted logarithmic slope. headWhy the mode regime matters If instead k=π/h moves with the grid, the ratio λ_k,h/k^2=4/π^2 stays constant, not one. The bound's factor k^4 warns that it is not uniform in arbitrary moving high modes. The displayed ratio…
Exact mathematical passage
\lambda_{k,h}/k^2=4/\pi^2
\pi^4Use in source · 1 passage
Search aliases: π^4 pi
Use in source · Read passage · 02_strengthened.tex, line 2033
[Low mode dispersion in dimension d] For every centered mode k, 0 ≤ 4π^2absk^2-λ_N(k) ≤ frac4π^43N^2 ∑_j=1^dk_j^4.
Exact mathematical passage
0 \leq 4\pi^2\abs{k}^2-\lambda_N(k) \leq \frac{4\pi^4}{3N^2} \sum_{j=1}^{d}k_j^4.
\Pi^{\mathrm{phys}}Use in source · 4 passages
Search aliases: Π^phys Pi mathrm
Use in source · Read passage · 01_finite_core.tex, line 1198
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\Pi^{\mathrm{phys}}
\pi_\lambdaUse in source · 2 passages
Search aliases: π_λ pi lambda
Use in source · Read passage · 02_strengthened.tex, line 242
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
\pi_\lambda:G\to U(d_\lambda)
\pi_\muUse in source · 2 passages
Search aliases: π_μ pi mu
Use in source · Read passage · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
\Pi_\ell^{\mathrm{phys}}Use in source · 1 passage
Search aliases: Π_ℓ^phys Pi ell mathrm
Use in source · Read passage · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\rho_+Use in source · 2 passages
Search aliases: ρ_+ rho
Use in source · monograph_atlas.tex, line 320
For ρ_+=frac12mat1 · 1 1 · 1, the exact output is Φ(ρ_+)=mat17/25 · 2/5
Exact mathematical passage
\rho_+=\frac12\mat{1&1\\1&1}
\rho_0Use in source · 2 passages
Search aliases: ρ_0 rho
Use in source · Read passage · 02_strengthened.tex, line 1385
The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.
Exact mathematical passage
\rho_0
\rho_{00}Use in source · 1 passage
Search aliases: ρ_00 rho
Use in source · monograph_atlas.tex, line 318
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
(\rho_{00},\rho_{01},\rho_{10},\rho_{11})
\rho_{01}Use in source · 1 passage
Search aliases: ρ_01 rho
Use in source · monograph_atlas.tex, line 318
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
(\rho_{00},\rho_{01},\rho_{10},\rho_{11})
\rho_{10}Use in source · 1 passage
Search aliases: ρ_10 rho
Use in source · monograph_atlas.tex, line 318
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
(\rho_{00},\rho_{01},\rho_{10},\rho_{11})
\rho_{11}Use in source · 1 passage
Search aliases: ρ_11 rho
Use in source · monograph_atlas.tex, line 318
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
(\rho_{00},\rho_{01},\rho_{10},\rho_{11})
\rho_{\rm Bell}Use in source · 3 passages
Search aliases: ρ_rm Bell rho rm
Use in source · monograph_atlas.tex, line 333
lassical input II.7.7. Ordinary transpose T sends a positive matrix on one system to another positive matrix and preserves its trace. Those properties are insufficient for a quantum channel on a subsystem of an entangled state. The test must include id⊗ T on the larger system. vfigcomplete_positivity Apply this extended map to the Bell density matrix. The result is (id⊗ T)(ρ_rm Bell) =frac12mat1 · 0 · 0 · 0 0 · 0 · 1 · 0 0 · 1 · 0 · 0 0 · 0 · 0 · 1. Its middle block has one negative eigenvalue. …
Exact mathematical passage
(\id\otimes\mathsf T)(\rho_{\rm Bell}) =\frac12\mat{1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1}.
\rho_\etaUse in source · 1 passage
Search aliases: ρ_η rho eta
Use in source · monograph_atlas.tex, line 503
actual amplitude vector is (- i,1, i,1)^T/2, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state. headA channel changes the conclusion predictably Before evolving, mix the seed density matrix with its complete momentum dephasing: ρ_η=(1-η)ket0bra0+η∑_jP_jket0bra0P_j =(1-η)ket0bra0+η I/4, quad0≤η≤1. At transfer time, the probability at th…
Exact mathematical passage
\rho_\eta=(1-\eta)\ket0\bra0+\eta\sum_jP_j\ket0\bra0P_j =(1-\eta)\ket0\bra0+\eta I/4, \quad0\le\eta\le1.
\bm\sigmaUse in source · 2 passages
Search aliases: σ bm sigma
Use in source · Read passage · 02_strengthened.tex, line 891
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
P(\theta,\phi) = \frac12\left(I+\bm n(\theta,\phi)\cdot\bm\sigma\right),
\boldsymbol\sigmaUse in source · 3 passages
Search aliases: σ boldsymbol sigma
Use in source · Read passage · 01_finite_core.tex, line 637
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
P(\theta,\phi) = \frac12\left(I+\mathbf n(\theta,\phi)\cdot\boldsymbol\sigma\right),
\Sigma'Use in source · 2 passages
Search aliases: Σ' Sigma
Use in source · Read passage · 02_strengthened.tex, line 1608
[Uniform self energy bounds] Let δ=dist(z,Spec D)>0, g=norm B. Then normΣ(z)≤fracg^2δ and normΣ'(z)≤fracg^2δ^2. For real E,E' in an interval at distance at least δ from Spec D, normΣ(E)-Σ(E') ≤ fracg^2δ^2absE-E'.
Exact mathematical passage
\norm{\Sigma'(z)}\leq\frac{g^2}{\delta^2}.
\sigma_aUse in source · 2 passages
Search aliases: σ_a sigma
Use in source · monograph_atlas.tex, line 746
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical ke…
Exact mathematical passage
C_a=0\oplus\sigma_a
\Sigma_PUse in source · 2 passages
Search aliases: Σ_P Sigma
Use in source · Read passage · 01_finite_core.tex, line 937
The self-energy Σ_P(z) =H_PQ(z-H_QQ)^-1H_QP contains the complete influence of the eliminated packet. Bare truncation drops Σ_P(z) and is therefore exact only when the coupling between packets vanishes or the self-energy cancels for another stated reason.
Exact mathematical passage
\Sigma_P(z) =H_{PQ}(z-H_{QQ})^{-1}H_{QP}
\phiUse in source · 21 passages
Search aliases: ϕ phi
Use in source · monograph_atlas.tex, line 272
vpage15Curvature measures a local area responseAN EXACT BLOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away…
Exact mathematical passage
n=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)
\Phi^{(J)}Use in source · 2 passages
Search aliases: Φ^(J) Phi
Use in source · Read passage · 03_supplement_02.tex, line 206
To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive …
Exact mathematical passage
[\Phi^{(J)}(X)]_{(i,u),(j,v)} =\Phi\bigl(X_{(i,\cdot),(j,\cdot)}\bigr)_{u,v}.
\Phi^+Use in source · 1 passage
Search aliases: Φ^+ Phi
Use in source · monograph_atlas.tex, line 294
vpage16Which coherence can a measurement see?THE ANSWER DEPENDS ON ITS EFFECTSTraditional edition: Theorem II.7.2, Theorem II.7.4, Corollary II.7.5. For the Bell density matrix ρ=ketΦ^+braΦ^+, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses 00 and 11. vfigdephasing Every effect in the computational basis E_a=ket abra a has the same expectation on both matrices, namely 1/2 at 00 and…
Exact mathematical passage
\rho=\ket{\Phi^+}\bra{\Phi^+}
\Phi_{cd,ab}Use in source · 3 passages
Search aliases: Φ_cd,ab Phi
Use in source · Read passage · 02_strengthened.tex, line 1043
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
\Phi_{cd,ab} = \Qcal_{cd}\Phi\Pcal_{ab}.
\Phi_{\alpha\beta;\gamma\delta}Use in source · 1 passage
Search aliases: Φ_αβ;γδ Phi alpha beta gamma delta
Use in source · Read passage · 01_finite_core.tex, line 751
Packet superoperator blocks are defined by Φ_αβ;γδ(X) = P_α Φ(P_γ XP_δ) P_β. Summing all four packet indices reconstructs Φ(X) exactly.
Exact mathematical passage
\Phi_{\alpha\beta;\gamma\delta}(X) = P_\alpha \Phi(P_\gamma XP_\delta) P_\beta.
\Phi_\gammaUse in source · 1 passage
Search aliases: Φ_γ Phi gamma
Use in source · Read passage · 02_strengthened.tex, line 1196
0 · 0 . The channel Φ_γ(ρ)=K_0ρ K_0^*+K_1ρ K_1^* is trace preserving. It sends the off diagonal packet ket0bra1 to sqrt1-γket0bra1, while the excited population transfers partly into the ground population.
Exact mathematical passage
\Phi_\gamma(\rho)=K_0\rho K_0^*+K_1\rho K_1^*
\PsiUse in source · 4 passages
Search aliases: Ψ Psi
Use in source · Read passage · 01_finite_core.tex, line 1239
[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.
Exact mathematical passage
\Psi_{g,\ell}
\psi^*Use in source · 5 passages
Search aliases: ψ^* psi
Use in source · monograph_atlas.tex, line 10
to follow when its objects remain visible. This appendix follows a state vector, a density matrix, and a retained part of a dynamical system. They may describe the same experiment, but they do not answer the same question. vfigthree_objects headAmplitudes retain relative phase For the Bell vector ψ=(ket00+ket11)/sqrt2, the two nonzero entries are amplitudes. The density matrix ρ=ψψ^* also records coherence in its off-diagonal entries. Measurements that compare the two addresses can detect this c…
Exact mathematical passage
\rho=\psi\psi^*
\Psi_{ef,cd}Use in source · 1 passage
Search aliases: Ψ_ef,cd Psi
Use in source · Read passage · 02_strengthened.tex, line 1055
[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.
Exact mathematical passage
(\Psi\Phi)_{ef,ab} = \sum_{c,d} \Psi_{ef,cd}\Phi_{cd,ab}.
\Psi_{g,\ell}Use in source · 1 passage
Search aliases: Ψ_g,ℓ Psi ell
Use in source · Read passage · 01_finite_core.tex, line 1239
[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.
Exact mathematical passage
\Psi_{g,\ell}
\psi_iUse in source · 1 passage
Search aliases: ψ_i psi
Use in source · Read passage · 01_finite_core.tex, line 92
A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\psi_i\in\bbC^{d_i}\otimes\bbC^{m_i}
\OmegaUse in source · 6 passages
Search aliases: Ω Omega
Use in source · Read passage · 02_strengthened.tex, line 1158
Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.
Exact mathematical passage
\Omega=\sum_i\ket{i}\otimes\ket{i}.
\omega^{(k-j)r}Use in source · 1 passage
Search aliases: ω^(k-j)r omega
Use in source · Read passage · 01_finite_core.tex, line 217
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
\frac1N\sum_{r=0}^{N-1}\omega^{(k-j)r}=\delta_{jk},
\omega^{-jr}Use in source · 2 passages
Search aliases: ω^-jr omega
Use in source · Read passage · 01_finite_core.tex, line 196
[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.
Exact mathematical passage
\boxed{ P_j = \frac1N\sum_{r=0}^{N-1}\omega^{-jr}S^r, \qquad 0\le j<N, }
\omega^{-qr}Use in source · 1 passage
Search aliases: ω^-qr omega
Use in source · Read passage · 02_strengthened.tex, line 1099
[Coherence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.
Exact mathematical passage
\Lcal_q(X) = \frac1N \sum_{r=0}^{N-1} \omega^{-qr}S^rXS^{-r}.
\omega^jUse in source · 4 passages
Search aliases: ω^j omega
Use in source · Read passage · 01_finite_core.tex, line 206
[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.
Exact mathematical passage
P_jP_k=\delta_{jk}P_j, \qquad \sum_{j=0}^{N-1}P_j=I, \qquad SP_j=\omega^jP_j.
\omega^{jr}Use in source · 2 passages
Search aliases: ω^jr omega
Use in source · Read passage · 01_finite_core.tex, line 221
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
P_j^*=\frac1N\sum_{r=0}^{N-1}\omega^{jr}S^{-r}=P_j.
\omega^{q+j}Use in source · 1 passage
Search aliases: ω^q+j omega
Use in source · Read passage · 02_strengthened.tex, line 313
For x∈ P_j H, S(Ax) =(SAS^*)Sx =ω^q+jAx. Thus Ax∈ P_j+q H. The block statement follows.
Exact mathematical passage
S(Ax) =(SAS^*)Sx =\omega^{q+j}Ax.
\omega^{q_r}Use in source · 1 passage
Search aliases: ω^q_r omega
Use in source · Read passage · 02_strengthened.tex, line 1237
[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.
Exact mathematical passage
SL_rS^*=\omega^{q_r}L_r.
\omega^{\operatorname{val}(r)}Use in source · 1 passage
Search aliases: ω^val(r) omega operatorname
Use in source · Read passage · 03_supplement_03.tex, line 93
Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N is proved from S^N=I, giving ρ(r+s)=ρ(r)ρ(s) and ρ(0)=I. The final matrix theorem derives the representation law …
Exact mathematical passage
\omega=\exp(2\pi i/N),\qquad \chi(r)=\omega^{\operatorname{val}(r)}.
\omega^{\operatorname{val}(r)\operatorname{val}(j)}Use in source · 1 passage
Search aliases: ω^val(r)val(j) omega operatorname operatorname
Use in source · Read passage · 03_supplement_03.tex, line 109
Exact coefficient formula. The packet definition is a finite sum over residue classes: P_j=frac1N∑_r∈ Z/N Zχ(-rj)ρ(r). The source proves χ(-rj)=(ω^val(r)val(j))^-1, so this is precisely the manuscript's N-term Fourier projector formula, with canonical representatives.
Exact mathematical passage
\chi(-rj)=\bigl(\omega^{\operatorname{val}(r)\operatorname{val}(j)}\bigr)^{-1},
\ell^2Use in source · 2 passages
Search aliases: ℓ^2 ell
Use in source · monograph_atlas.tex, line 769
a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and give a fixed complet…
Exact mathematical passage
\ell^2(\NN_0)
\toUse in source · 31 passages
Search aliases: → to
Use in source · monograph_atlas.tex, line 452
ues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect the finite size of the system As τ→0, every eigenmode contributes approximately one and the dimension diagnostic tends to zero. As τ→∞, the zero mode dominates, so it again tends to zero. A finite spectrum cann…
Exact mathematical passage
\tau\to0
\mapstoUse in source · 8 passages
Search aliases: ↦ mapsto
Use in source · monograph_atlas.tex, line 580
vpage31Coherence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSTraditional edition: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space are L_0(X)=(X+ZXZ)/2 and L_1(X)=(X-ZXZ)/2.
Exact mathematical passage
X\mapsto ZXZ
\RightarrowUse in source · 1 passage
Search aliases: ⇒ Rightarrow
Use in source · Read passage · 03_supplement_05.tex, line 101
a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzero p, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication X^*Xv=0⇒ Xv=0 then gives the required proportionality. When p=0, the Gram identity forces X=0 directly.
Exact mathematical passage
X^*Xv=0\Rightarrow Xv=0
\forallUse in source · 1 passage
Search aliases: ∀ forall
Use in source · Read passage · 03_supplement_03.tex, line 18
Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and proves the equivalence the seeds generate Longleftrightarrow rank D=m. Thus m≤ kd is necessary. It is sufficient as well: embed the m coordinate columns i…
Exact mathematical passage
\forall X\in M_{d\times m}(\C),\quad \exists A_1,\ldots,A_k\in M_d(\C):\quad X=\sum_{a=1}^k A_aC_a.
\partialUse in source · 6 passages
Search aliases: ∂ partial
Use in source · monograph_atlas.tex, line 277
LOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T.…
Exact mathematical passage
F_{\theta\phi}=-\ii\Tr\!\left(P[\partial_\theta P,\partial_\phi P]\right) =\frac12\sin\theta.
\partial_\thetaUse in source · 5 passages
Search aliases: ∂_θ partial theta
Use in source · monograph_atlas.tex, line 277
LOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T.…
Exact mathematical passage
F_{\theta\phi}=-\ii\Tr\!\left(P[\partial_\theta P,\partial_\phi P]\right) =\frac12\sin\theta.
\partial_\muUse in source · 1 passage
Search aliases: ∂_μ partial mu
Use in source · Read passage · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\partial_\nuUse in source · 1 passage
Search aliases: ∂_ν partial nu
Use in source · Read passage · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\partial_\phiUse in source · 5 passages
Search aliases: ∂_ϕ partial phi
Use in source · monograph_atlas.tex, line 277
LOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T.…
Exact mathematical passage
F_{\theta\phi}=-\ii\Tr\!\left(P[\partial_\theta P,\partial_\phi P]\right) =\frac12\sin\theta.
\existsUse in source · 2 passages
Search aliases: ∃ exists
Use in source · Read passage · 03_supplement_03.tex, line 18
Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and proves the equivalence the seeds generate Longleftrightarrow rank D=m. Thus m≤ kd is necessary. It is sufficient as well: embed the m coordinate columns i…
Exact mathematical passage
\forall X\in M_{d\times m}(\C),\quad \exists A_1,\ldots,A_k\in M_d(\C):\quad X=\sum_{a=1}^k A_aC_a.
\nabla^{(a)}_XUse in source · 2 passages
Search aliases: ∇^(a)_X nabla
Use in source · Read passage · 03_supplement_06.tex, line 88
The local connection and its change of coordinates. For a coordinate function u valued in E_a, define ∇^(a)_Xu=A_a(P D_M(B_a u)[X]). For every overlap point, Lean proves the operator identity ∇^(b)_X(g_abu)=g_ab∇^(a)_Xu. This is proved for arbitrary local coordinate functions. It does not assume that u was obtained from a globally smooth section or that a smooth extension has been supplied. Together with the local smoothness criterion, this closes the gluing step left open in the fifth block.
Exact mathematical passage
\nabla^{(a)}_Xu=A_a\bigl(P\,D_M(B_a u)[X]\bigr).
\nabla^{(b)}_XUse in source · 1 passage
Search aliases: ∇^(b)_X nabla
Use in source · Read passage · 03_supplement_06.tex, line 92
The local connection and its change of coordinates. For a coordinate function u valued in E_a, define ∇^(a)_Xu=A_a(P D_M(B_a u)[X]). For every overlap point, Lean proves the operator identity ∇^(b)_X(g_abu)=g_ab∇^(a)_Xu. This is proved for arbitrary local coordinate functions. It does not assume that u was obtained from a globally smooth section or that a smooth extension has been supplied. Together with the local smoothness criterion, this closes the gluing step left open in the fifth block.
Exact mathematical passage
\boxed{\nabla^{(b)}_X(g_{ab}u)=g_{ab}\nabla^{(a)}_Xu.}
\nabla_{[X,Y]}Use in source · 3 passages
Search aliases: ∇_[X,Y] nabla
Use in source · Read passage · 01_finite_core.tex, line 580
The rank of P is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section s, so that Ps=s, define ∇_Xs=P ds(X). This is a connection on smooth range sections. For smooth vector fields X,Y, its curvature satisfies (∇_X∇_Y-∇_Y∇_X-∇_[X,Y])s =P[dP(X),dP(Y)]P s. Equivalently, its endomorphism-valued curvature is F=P(dPwedge dP)P. No global frame or globally constant rank is required, and the vector fields need not commute. The geometric formalizat…
Exact mathematical passage
\boxed{ (\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]})s =P[dP(X),dP(Y)]P\,s. }
\nabla_{fX}Use in source · 1 passage
Search aliases: ∇_fX nabla
Use in source · Read passage · 01_finite_core.tex, line 606
textitConnection rules. Smoothness of P ds(X) and P^2=P show that ∇_Xs is a smooth range section. Differentiation and projection give ∇_X+Ys · =∇_Xs+∇_Ys, · ∇_fXs · =f∇_Xs, ∇_X(s+t) · =∇_Xs+∇_Xt, · ∇_X(fs) · =f∇_Xs+(Xf)s for smooth real f. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bundle.
Exact mathematical passage
\begin{aligned} \nabla_{X+Y}s&=\nabla_Xs+\nabla_Ys,& \nabla_{fX}s&=f\nabla_Xs,\\ \nabla_X(s+t)&=\nabla_Xs+\nabla_Xt,& \nabla_X(fs)&=f\nabla_Xs+(Xf)s \end{aligned}
\nabla_{X+Y}Use in source · 1 passage
Search aliases: ∇_X+Y nabla
Use in source · Read passage · 01_finite_core.tex, line 606
textitConnection rules. Smoothness of P ds(X) and P^2=P show that ∇_Xs is a smooth range section. Differentiation and projection give ∇_X+Ys · =∇_Xs+∇_Ys, · ∇_fXs · =f∇_Xs, ∇_X(s+t) · =∇_Xs+∇_Xt, · ∇_X(fs) · =f∇_Xs+(Xf)s for smooth real f. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bundle.
Exact mathematical passage
\begin{aligned} \nabla_{X+Y}s&=\nabla_Xs+\nabla_Ys,& \nabla_{fX}s&=f\nabla_Xs,\\ \nabla_X(s+t)&=\nabla_Xs+\nabla_Xt,& \nabla_X(fs)&=f\nabla_Xs+(Xf)s \end{aligned}
\nabla_YUse in source · 4 passages
Search aliases: ∇_Y nabla
Use in source · Read passage · 01_finite_core.tex, line 580
The rank of P is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section s, so that Ps=s, define ∇_Xs=P ds(X). This is a connection on smooth range sections. For smooth vector fields X,Y, its curvature satisfies (∇_X∇_Y-∇_Y∇_X-∇_[X,Y])s =P[dP(X),dP(Y)]P s. Equivalently, its endomorphism-valued curvature is F=P(dPwedge dP)P. No global frame or globally constant rank is required, and the vector fields need not commute. The geometric formalizat…
Exact mathematical passage
\boxed{ (\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]})s =P[dP(X),dP(Y)]P\,s. }
\prodUse in source · 2 passages
Search aliases: ∏ prod
Use in source · Read passage · 03_supplement_02.tex, line 21
For every binary address a, W_a=∏_j=0^n-1X_j^a_j=T_a, W_aψ_0=ket a. The family (W_aψ_0)_a∈ A_n is linearly independent, spans V_n, and has exactly 2^n elements. It is the standard computational basis.
Exact mathematical passage
W_a=\prod_{j=0}^{n-1}X_j^{a_j}=T_a, \qquad W_a\psi_0=\ket a.
\prod_{i\in F}Use in source · 1 passage
Search aliases: ∏_i∈ F prod in
Use in source · Read passage · 03_supplement_02.tex, line 48
Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite number of factors with the stated finite coordinate sets.
Exact mathematical passage
\mathsf T(A)(x,y)=\prod_{i\in F} A_i(x_i,y_i).
\prod_{j=0}^{n-1}Use in source · 1 passage
Search aliases: ∏_j=0^n-1 prod
Use in source · Read passage · 03_supplement_02.tex, line 21
For every binary address a, W_a=∏_j=0^n-1X_j^a_j=T_a, W_aψ_0=ket a. The family (W_aψ_0)_a∈ A_n is linearly independent, spans V_n, and has exactly 2^n elements. It is the standard computational basis.
Exact mathematical passage
W_a=\prod_{j=0}^{n-1}X_j^{a_j}=T_a, \qquad W_a\psi_0=\ket a.
\sum_{a,b}Use in source · 5 passages
Search aliases: ∑_a,b sum
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
\sum_{a,b,c}Use in source · 1 passage
Search aliases: ∑_a,b,c sum
Use in source · monograph_atlas.tex, line 136
le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original 4×4 circuit matrix: ∑_a,b,cP_b CNOT P_a (H_rm Had⊗ I)P_c =CNOT(H_rm Had⊗ I). This expansion is an identity even though the intermediate parity …
Exact mathematical passage
\sum_{a,b,c}P_b\,\mathrm{CNOT}\,P_a (H_{\rm Had}\otimes I)P_c =\mathrm{CNOT}(H_{\rm Had}\otimes I).
\sum_{a,b,c,d}Use in source · 1 passage
Search aliases: ∑_a,b,c,d sum
Use in source · Read passage · 02_strengthened.tex, line 1051
[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.
Exact mathematical passage
\Phi=\sum_{a,b,c,d}\Phi_{cd,ab}.
\sum_{a:Z_{L,a}(t)>0}Use in source · 1 passage
Search aliases: ∑_a:Z_L,a(t)>0 sum
Use in source · Read passage · 02_strengthened.tex, line 1999
[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.
Exact mathematical passage
d_L(t) = \sum_{a:Z_{L,a}(t)>0} \frac{Z_{L,a}(t)}{Z_L(t)} d_{L,a}(t),
\sum_{a=0}^1Use in source · 1 passage
Search aliases: ∑_a=0^1 sum
Use in source · monograph_atlas.tex, line 556
vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILURETraditional edition: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5 4/5 · 3/5, Q_a=RP_aR^*, S=∑_a=0^1Q_aP_a=frac35R. The exact Gram matrices give (S-I)^*(S-I)=frac1625I, S^*S=frac925I, normS-I=frac45<1. Consequently the unitary polar factor is S(S^*S)^-1/2=R, and Q_aR=RP_a for both complete packets. xfigpolar headThe norm condition selects a chart If the target rank-one…
Exact mathematical passage
R=\mat{3/5&-4/5\\4/5&3/5},\qquad Q_a=RP_aR^*, \qquad S=\sum_{a=0}^1Q_aP_a=\frac35R.
\sum_{a=1}^kUse in source · 1 passage
Search aliases: ∑_a=1^k sum
Use in source · Read passage · 03_supplement_03.tex, line 18
Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and proves the equivalence the seeds generate Longleftrightarrow rank D=m. Thus m≤ kd is necessary. It is sufficient as well: embed the m coordinate columns i…
Exact mathematical passage
\forall X\in M_{d\times m}(\C),\quad \exists A_1,\ldots,A_k\in M_d(\C):\quad X=\sum_{a=1}^k A_aC_a.
\sum_{a=1}^{s}Use in source · 1 passage
Search aliases: ∑_a=1^s sum
Use in source · Read passage · 03_supplement_03.tex, line 53
An independent definition of Schmidt rank. The source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length s is the concrete equality C_ij=∑_a=1^su_a(i)v_a(j). It factors C=UV through an s-dimensional intermediate space, so rank C≤ s. Conversely, choose a basis of the row space of C. Expanding each row in this basis constructs a decomposition with exactly rank C product terms. Thus r(ψ)=rank C. The definition uses a least natural number, and th…
Exact mathematical passage
C_{ij}=\sum_{a=1}^{s}u_a(i)v_a(j).
\sum_{a_1,\ldots,a_{m-1}}Use in source · 2 passages
Search aliases: ∑_a_1,…,a_m-1 sum ldots
Use in source · Read passage · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
\sum_{a\in I}Use in source · 2 passages
Search aliases: ∑_a∈ I sum in
Use in source · Read passage · 02_strengthened.tex, line 6
The elementary coordinate object is a complete family of orthogonal projectors P_aP_b=δ_abP_a, ∑_a∈ IP_a=I. Every vector has packet coordinates P_aψ. Every operator has transfer blocks A_ba=P_bAP_a. These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction, effective dynamics, and geometric limits.
Exact mathematical passage
P_aP_b=\delta_{ab}P_a, \qquad \sum_{a\in I}P_a=I.
\sum_bUse in source · 6 passages
Search aliases: ∑_b sum
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
\sum_{b,a}Use in source · 1 passage
Search aliases: ∑_b,a sum
Use in source · Read passage · 03_supplement_01.tex, line 12
0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.
Exact mathematical passage
\sum_{b,a}U_{ba}=U,\qquad (UV)_{ba}=\sum_c U_{bc}V_{ca}.
\sum_CUse in source · 1 passage
Search aliases: ∑_C sum
Use in source · Read passage · 01_finite_core.tex, line 1199
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\boxed{ \mathfrak A_{AB} = [\widehat C_A,\widehat C_B] -i\sum_C f_{AB}^{\ \ C}\widehat C_C. }
\sum_{c,d}Use in source · 1 passage
Search aliases: ∑_c,d sum
Use in source · Read passage · 02_strengthened.tex, line 1055
[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.
Exact mathematical passage
(\Psi\Phi)_{ef,ab} = \sum_{c,d} \Psi_{ef,cd}\Phi_{cd,ab}.
\sum_{g\in G}Use in source · 2 passages
Search aliases: ∑_g∈ G sum in
Use in source · Read passage · 02_strengthened.tex, line 207
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda = \frac{d_\lambda}{\abs{G}} \sum_{g\in G} \overline{\chi_\lambda(g)}U(g)
\sum_{g\in S_3}Use in source · 1 passage
Search aliases: ∑_g∈ S_3 sum in
Use in source · monograph_atlas.tex, line 534
, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and three-cycle classes are c|rrr|r · e · (12) · (123) · d_λ trivial · 1 · 1 · 1 · 1 sign · 1 · -1 · 1 · 1 standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3…
Exact mathematical passage
\begin{array}{c|rrr|r} & e & (12)&(123)&d_\lambda\\\hline \text{trivial}&1&1&1&1\\ \text{sign}&1&-1&1&1\\ \text{standard}&2&0&-1&2 \end{array} \qquad P_\lambda=\frac{d_\lambda}{6}\sum_{g\in S_3}\chi_\lambda(g^{-1})U(g).
\sum_{i,j}Use in source · 1 passage
Search aliases: ∑_i,j sum
Use in source · Read passage · 02_strengthened.tex, line 1077
[Finite Kraus and Choi criteria] A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.
Exact mathematical passage
J(\Phi) = \sum_{i,j}\Phi(\ket{i}\bra{j})\otimes\ket{i}\bra{j}
\sum_{i=1}^{d_\lambda}Use in source · 1 passage
Search aliases: ∑_i=1^d_λ sum lambda
Use in source · Read passage · 02_strengthened.tex, line 259
x an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
P_\lambda=\sum_{i=1}^{d_\lambda}E_{ii}^{\lambda}.
\sum_jUse in source · 20 passages
Search aliases: ∑_j sum
Use in source · monograph_atlas.tex, line 102
Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.
Exact mathematical passage
\sum_jP_j=I
\sum_{j,k}Use in source · 5 passages
Search aliases: ∑_j,k sum
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
\sum_{j,k=1}^{m}Use in source · 1 passage
Search aliases: ∑_j,k=1^m sum
Use in source · Read passage · 02_strengthened.tex, line 1714
[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.
Exact mathematical passage
\sum_{j,k=1}^{m} \ip{v_j}{K(t_j-t_k)v_k} \geq0.
\sum_{j=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_j=0^N-1 sum
Use in source · Read passage · 01_finite_core.tex, line 206
[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.
Exact mathematical passage
P_jP_k=\delta_{jk}P_j, \qquad \sum_{j=0}^{N-1}P_j=I, \qquad SP_j=\omega^jP_j.
\sum_{j=1}^{d}Use in source · 2 passages
Search aliases: ∑_j=1^d sum
Use in source · Read passage · 02_strengthened.tex, line 2022
Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].
Exact mathematical passage
\lambda_N(k) = 4N^2 \sum_{j=1}^{d} \sin^2\left(\frac{\pi k_j}{N}\right),
\sum_{j_1+\cdots+j_m=r}Use in source · 1 passage
Search aliases: ∑_j_1+⋯+j_m=r sum cdots
Use in source · Read passage · 02_strengthened.tex, line 351
[Tensor charge refinement] For r∈ Z/N Z, define P_r^rm total = ∑_j_1+⋯+j_m=r P_j_1^(1)⊗⋯⊗ P_j_m^(m). Then the P_r^rm total form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.
Exact mathematical passage
P_r^{\rm total} = \sum_{j_1+\cdots+j_m=r} P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
\sum_{j_1+\cdots+j_m\equiv r\pmod N}Use in source · 1 passage
Search aliases: ∑_j_1+⋯+j_m≡ rpmod N sum cdots equiv pmod
Use in source · Read passage · 01_finite_core.tex, line 247
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
\boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. }
\sum_kUse in source · 5 passages
Search aliases: ∑_k sum
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
\sum_{k=1}^{m}Use in source · 1 passage
Search aliases: ∑_k=1^m sum
Use in source · Read passage · 02_strengthened.tex, line 1723
Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.
Exact mathematical passage
w=\sum_{k=1}^{m}\e^{\ii t_kD}B^*v_k.
\sum_{k\in\ZZ^d}Use in source · 3 passages
Search aliases: ∑_k∈ZZ^d ∑_k∈ Z^d sum in ZZ sum in mathbb
Use in source · Read passage · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
\sum_{k\in\Lambda_N^*}Use in source · 2 passages
Search aliases: ∑_k∈Λ_N^* sum in Lambda
Use in source · Read passage · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
\sum_{n=0}^{\infty}Use in source · 4 passages
Search aliases: ∑_n=0^∞ sum infty
Use in source · monograph_atlas.tex, line 670
|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings. headA series near zero energy and its exact remainder At z=1/2, |z| |D^-1|=1/4<1, so Σ(z)=-∑_n=0^∞fracz^n2^n+1. After the first four terms, the exact remainder has magnitude 1/384. The geometric series controls the infinite tail. The retained che…
Exact mathematical passage
\Sigma(z)=-\sum_{n=0}^{\infty}\frac{z^n}{2^{n+1}}.
\sum_{n\in\ZZ^d}Use in source · 1 passage
Search aliases: ∑_n∈ZZ^d ∑_n∈ Z^d sum in ZZ sum in mathbb
Use in source · Read passage · 02_strengthened.tex, line 2107
[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.
Exact mathematical passage
Z_d(t) = \sum_{k\in\ZZ^d}\e^{-4\pi^2t\abs{k}^2} = (4\pi t)^{-d/2} \sum_{n\in\ZZ^d}\e^{-\abs{n}^2/(4t)}.
\sum_{n\neq0}Use in source · 1 passage
Search aliases: ∑_n≠0 sum neq
Use in source · Read passage · 02_strengthened.tex, line 2140
By [source reference], Z_d(t) =(4π t)^-d/2(1+R(t)), where R(t)=∑_n≠0 e^-absn^2/(4t) =O( e^-1/(4t)). Differentiation gives tR'(t)=O(t^-1 e^-1/(4t)). Substituting this estimate into the logarithmic derivative gives the stated small time expansion.
Exact mathematical passage
R(t)=\sum_{n\neq0}\e^{-\abs{n}^2/(4t)} =O\bigl(\e^{-1/(4t)}\bigr).
\sum_qUse in source · 1 passage
Search aliases: ∑_q sum
Use in source · Read passage · 02_strengthened.tex, line 485
[Charge pruning of circuit paths] Suppose every gate has a finite charge decomposition U_r=∑_qU_r^(q), SU_r^(q)S^*=ω^qU_r^(q). A term with gate charges q_1,…,q_m can contribute from source charge a_0 only to target charge a_m=a_0+q_1+⋯+q_m. Every packet path violating one of the intermediate charge updates vanishes.
Exact mathematical passage
U_r=\sum_qU_r^{(q)}, \qquad SU_r^{(q)}S^*=\omega^qU_r^{(q)}.
\sum_rUse in source · 12 passages
Search aliases: ∑_r sum
Use in source · Read passage · 01_finite_core.tex, line 855
_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP from the F_r. Every recovery branch of a coherent combination is therefore scala…
Exact mathematical passage
\sum_rK_r^*K_r+K_{\mathrm{unused}}^*K_{\mathrm{unused}} =S+(I-S)^2=I.
\sum_{r}Use in source · 1 passage
Search aliases: ∑_r sum
Use in source · Read passage · 02_strengthened.tex, line 1740
[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.
Exact mathematical passage
D=\sum_{r}E_rQ_r
\sum_{r,a}Use in source · 1 passage
Search aliases: ∑_r,a sum
Use in source · Read passage · 03_supplement_05.tex, line 91
Deriving the Gram matrix from recovery Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code implies ∑_r,a A_raρ A_ra^*=Pρ P. This equality on all matrices follows by applying correction to the sandwich supported on the code, Pρ P, using the p…
Exact mathematical passage
\sum_{r,a} A_{ra}\rho A_{ra}^*=P\rho P.
\sum_{r=0}^3Use in source · 1 passage
Search aliases: ∑_r=0^3 sum
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
\sum_{r=0}^{N-1}Use in source · 5 passages
Search aliases: ∑_r=0^N-1 sum
Use in source · Read passage · 01_finite_core.tex, line 196
[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.
Exact mathematical passage
\boxed{ P_j = \frac1N\sum_{r=0}^{N-1}\omega^{-jr}S^r, \qquad 0\le j<N, }
\sum_{r\in\mathbb Z/N\mathbb Z}Use in source · 1 passage
Search aliases: ∑_r∈ Z/N Z sum in mathbb mathbb
Use in source · Read passage · 03_supplement_03.tex, line 105
Exact coefficient formula. The packet definition is a finite sum over residue classes: P_j=frac1N∑_r∈ Z/N Zχ(-rj)ρ(r). The source proves χ(-rj)=(ω^val(r)val(j))^-1, so this is precisely the manuscript's N-term Fourier projector formula, with canonical representatives.
Exact mathematical passage
P_j=\frac1N\sum_{r\in\mathbb Z/N\mathbb Z}\chi(-rj)\rho(r).
\sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}}Use in source · 1 passage
Search aliases: ∑_substackγ,γ'∈Γ_b γ<γ' sum substack gamma gamma in Gamma gamma gamma
Use in source · Read passage · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\sum_{\alpha_1,\ldots,\alpha_{m-1}}Use in source · 1 passage
Search aliases: ∑_α_1,…,α_m-1 sum alpha ldots alpha
Use in source · Read passage · 01_finite_core.tex, line 283
[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.
Exact mathematical passage
\boxed{ P_{\alpha_m}UP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}U_mP_{\alpha_{m-1}} \cdots P_{\alpha_1}U_1P_{\alpha_0}. }
\sum_\betaUse in source · 2 passages
Search aliases: ∑_β sum beta
Use in source · Read passage · 01_finite_core.tex, line 302
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\boxed{ \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma} = \delta_{\alpha\gamma}P_\alpha. }
\sum_{\beta\succ\alpha}Use in source · 1 passage
Search aliases: ∑_βsuccα sum beta succ alpha
Use in source · Read passage · 01_finite_core.tex, line 1026
A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Exact mathematical passage
\boxed{ P_{\ell,\alpha} = C_\ell \left( \sum_{\beta\succ\alpha}P_{\ell+1,\beta} \right) E_\ell. }
\sum_{\gamma\in\Gamma_b}Use in source · 3 passages
Search aliases: ∑_γ∈Γ_b sum gamma in Gamma
Use in source · Read passage · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\sum_{\lambda_j>0}Use in source · 4 passages
Search aliases: ∑_λ_j>0 sum lambda
Use in source · Read passage · 02_strengthened.tex, line 1357
[Exact recovery on the error span] Under the hypotheses of [source reference], define on the syndrome support Q=∑_λ_j>0Q_j the completely positive map R_0(X) = ∑_λ_j>0 V_j^*Q_jXQ_jV_j. It extends to a quantum channel R on End( H). If Φ(X)=∑_aE_aXE_a^* is trace preserving on code states, then RΦ(ρ)=ρ for every density matrix ρ=Pρ P.
Exact mathematical passage
Q=\sum_{\lambda_j>0}Q_j
\sum_{\lambda_r>0}Use in source · 1 passage
Search aliases: ∑_λ_r>0 sum lambda
Use in source · Read passage · 01_finite_core.tex, line 850
titSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S…
Exact mathematical passage
S=\sum_{\lambda_r>0}V_rV_r^*
\sum_{\lambda\in\widehat G}Use in source · 1 passage
Search aliases: ∑_λ∈widehat G sum lambda in widehat
Use in source · Read passage · 02_strengthened.tex, line 215
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda, \qquad \sum_{\lambda\in\widehat G}P_\lambda=I_{\Hcal}.
\sum_{\sum_i a_i=r}Use in source · 1 passage
Search aliases: ∑_∑_i a_i=r sum sum
Use in source · Read passage · 03_supplement_02.tex, line 54
Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.
Exact mathematical passage
Q_a=\mathsf T\bigl(i\mapsto P^{(i)}_{a_i}\bigr), \qquad C_r=\sum_{\sum_i a_i=r}Q_a.
\inftyUse in source · 28 passages
Search aliases: ∞ infty
Use in source · monograph_atlas.tex, line 452
τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect the finite size of the system As τ→0, every eigenmode contributes approximately one and the dimension diagnostic tends to zero. As τ→∞, the zero mode dominates, so it again tends to zero. A finite spectrum cannot sustain a nonzero plateau at all scales.
Exact mathematical passage
\tau\to\infty
\capUse in source · 1 passage
Search aliases: ∩ cap
Use in source · Read passage · 03_supplement_06.tex, line 49
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
U_a\cap U_b
\int_0^{2\pi}Use in source · 1 passage
Search aliases: ∫_0^2π int pi
Use in source · Read passage · 02_strengthened.tex, line 910
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
\frac1{2\pi} \int_0^{2\pi}\int_0^\pi \Fcal_{\theta\phi}\dd\theta\dd\phi =1.
\int_0^\piUse in source · 1 passage
Search aliases: ∫_0^π int pi
Use in source · Read passage · 02_strengthened.tex, line 910
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
\frac1{2\pi} \int_0^{2\pi}\int_0^\pi \Fcal_{\theta\phi}\dd\theta\dd\phi =1.
\int_0^\inftyUse in source · 1 passage
Search aliases: ∫_0^∞ int infty
Use in source · Read passage · 02_strengthened.tex, line 1692
· - i∫_0^tK(t-s)x(s) d s, where K(t)=B e^- i tDB^*. For Res>0, ∫_0^∞ e^-stK(t) d t = B(s+ i D)^-1B^*.
Exact mathematical passage
\int_0^\infty\e^{-st}K(t)\dd t = B(s+\ii D)^{-1}B^*.
\simeqUse in source · 4 passages
Search aliases: ≃ simeq
Use in source · Read passage · 03_supplement_03.tex, line 9
A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d× m coefficient space is equivalent to rank C=m. Consequently a cyclic seed requires m≤ d. The assumption that there is at least one row is essential …
Exact mathematical passage
\mathcal O(C)\simeq \operatorname{Row}(C)^d, \qquad \boxed{\dim_\C\mathcal O(C)=d\,\rank C.}
\congUse in source · 7 passages
Search aliases: ≅ cong
Use in source · Read passage · 01_finite_core.tex, line 79
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cA \cong \bigoplus_{i=1}^r \left(M_{d_i}(\bbC)\otimes I_{m_i}\right)
\neUse in source · 20 passages
Search aliases: ≠ ne
Use in source · monograph_atlas.tex, line 102
Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.
Exact mathematical passage
j\ne k
\neqUse in source · 3 passages
Search aliases: ≠ neq
Use in source · Read passage · 02_strengthened.tex, line 524
[Exact invariant packets] The following conditions are equivalent: item The Hamiltonian commutes with each projector: [H,P_a]=0 for every a. item Transfer blocks between distinct packets vanish: P_bHP_a=0 whenever a≠ b. item Every packet P_a H is invariant under e^- i tH for every real t. When they hold, H=∑_aP_aHP_a and e^- i tH=∑_aP_a e^- i tHP_a.
Exact mathematical passage
a\neq b
\equivUse in source · 1 passage
Search aliases: ≡ equiv
Use in source · Read passage · 01_finite_core.tex, line 247
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
\boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. }
\leqUse in source · 36 passages
Search aliases: ≤ leq
Use in source · monograph_atlas.tex, line 632
On a two-dimensional code, consider the overlap compression A=PE_a^*E_bP=diag(11/10,9/10). For instance, within the code take E_a=I and E_b=A. This declares an error family, not by itself a normalized noise channel. The optimal scalar compression has inf_c∈ CnormA-cI=frac110, c=1 attains it. Indeed, for every complex c, the triangle inequality gives frac15=|frac1110-frac910| ≤|frac1110-c|+|frac910-c|. Thus the larger of the two distances is at least 1/10, while both distances equal 1/10 at c=1. …
Exact mathematical passage
\frac15=\left|\frac{11}{10}-\frac9{10}\right| \leq\left|\frac{11}{10}-c\right|+\left|\frac9{10}-c\right|.
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · Read passage · 01_finite_core.tex, line 792
Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environment can distinguish logical states and exact correction is impossible.
Exact mathematical passage
\cC\subset\cH
\oplusUse in source · 7 passages
Search aliases: ⊕ oplus
Use in source · monograph_atlas.tex, line 746
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical ke…
Exact mathematical passage
\CC^3=\CC\ket0\oplus\CC^2
\bigoplus_{i=1}^rUse in source · 2 passages
Search aliases: ⊕_i=1^r bigoplus
Use in source · Read passage · 01_finite_core.tex, line 79
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cA \cong \bigoplus_{i=1}^r \left(M_{d_i}(\bbC)\otimes I_{m_i}\right)
\bigoplus_jUse in source · 1 passage
Search aliases: ⊕_j bigoplus
Use in source · Read passage · 02_strengthened.tex, line 1111
erence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.
Exact mathematical passage
\Ran\Lcal_q = \bigoplus_jP_{j+q}\End(\Hcal)P_j.
\bigoplus_{r=1}^{s}Use in source · 2 passages
Search aliases: ⊕_r=1^s bigoplus
Use in source · Read passage · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
\bigoplus_\muUse in source · 1 passage
Search aliases: ⊕_μ bigoplus mu
Use in source · Read passage · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
\bigoplus_{\mu\in\widehat G}Use in source · 1 passage
Search aliases: ⊕_μ∈widehat G bigoplus mu in widehat
Use in source · Read passage · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
\perpUse in source · 2 passages
Search aliases: ⊥ perp
Use in source · Read passage · 02_strengthened.tex, line 1385
The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.
Exact mathematical passage
Q^\perp=I-Q
\bigcapUse in source · 1 passage
Search aliases: ⋂ bigcap
Use in source · Read passage · 02_strengthened.tex, line 2216
[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.
Exact mathematical passage
\Hcal_{\phys}=\bigcap_a\Ker C_a
\bigcap_aUse in source · 1 passage
Search aliases: ⋂_a bigcap
Use in source · Read passage · 02_strengthened.tex, line 2216
[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.
Exact mathematical passage
\Hcal_{\phys}=\bigcap_a\Ker C_a
\langleUse in source · 5 passages
Search aliases: ⟨ langle
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
\rangleUse in source · 5 passages
Search aliases: ⟩ rangle
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
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- Paper_VII_Traditional_Final.pdf
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- exact_seed_support.pdf
- exact_feshbach_compression.pdf
- exact_channel_scale.pdf
- 01_quantum_seed_architecture.pdf
- 10_tensor_parity_refinement.pdf
- 02_bell_seed_probabilities.pdf
- 04_circuit_packet_paths.pdf
- 03_parity_block_hamiltonian.pdf
- 05_leakage_bound.pdf
- 08_moving_frame_connection.pdf
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- 16_amplitude_damping_channel.pdf
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- 06_feshbach_vs_truncation.pdf
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- 07_operator_packets.pdf
- 08_channel_coherence.pdf
- 09_qec_syndromes.pdf
- 10_feshbach_spectrum.pdf
- 11_memory_kernel.pdf
- 12_reflection_positivity.pdf
- 13_spectral_dimension.pdf
- 15_constraint_obstruction.pdf
- 14_continuum_tower.pdf
- 16_continuum_boundary.pdf