Paper VII: II. Effective dynamics and refinement towers

II.1 Introduction

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Quantum mechanics repeatedly uses the same operation. A Hilbert space is split into sectors, an operator is resolved between those sectors, and the pieces are propagated or recombined. Symmetry sectors, eigenspaces, syndrome spaces, adiabatic subspaces, and retained model spaces are familiar examples. This article treats them as instances of one finite packet calculus.

The elementary coordinate object is a complete family of orthogonal projectors

\[ P_aP_b=\delta_{ab}P_a, \qquad \sum_{a\in I}P_a=I. \]

Every vector has packet coordinates \(P_a\psi\). 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.

Quantum information and operator theory in finite dimensions provide the standard background [NielsenChuang2010], [Watrous2018], [Bhatia1997]. Finite group projectors come from character theory [Serre1977], [FultonHarris1991]. Moving spectral subspaces and geometric phase originate in the work of Kato, Berry, Wilczek, and Zee [Kato1950], [Kato1995], [Berry1984], [WilczekZee1984]. Completely positive maps and quantum dynamical semigroups are treated in [Stinespring1955], [Choi1975], [Kraus1983], [Lindblad1976]. Exact error correction is governed by the Knill Laflamme condition [KnillLaflamme1997], [SchumacherNielsen1996], [BenyKempfKribs2007]. Projection methods and hidden state memory are linked by the Feshbach, Nakajima, and Zwanzig constructions [Feshbach1958], [Nakajima1958], [Zwanzig1960], [BravyiDiVincenzoLoss2011].

The first part begins with quantum seeds. A state is cyclic only relative to a declared observable algebra. For an algebra in multiplicity form, the local orbit dimension is controlled by the rank of a coefficient matrix. This recovers the relation between Schmidt rank and the dimension reached by local operations. The finite group section then constructs central isotypic projectors. Cyclic symmetry gives a sharper charge selection rule. An operator with charge \(q\) can move a state only from packet \(j\) to packet \(j+q\).

A circuit is resolved by inserting the identity between successive gates. This produces a sum over packet paths. The target probability contains diagonal path terms and interference between distinct paths. Symmetry can remove entire path families before any matrix multiplication is performed. Unitarity gives exact row and column relations among the transfer blocks.

The moving packet section treats time dependent projectors. The canonical generator

\[ K(t)=\sum_a\dot P_a(t)P_a(t) \]

is skew adjoint and transports every packet. In the moving frame, the physical Hamiltonian acquires the geometric term \(-{\mathrm i} W(t)^*\dot W(t)\). The leakage estimate must account for the motion of the packet itself. The physical coupling \(QHP\) is therefore replaced by

\[ Q(t)\bigl(H(t)-{\mathrm i} K(t)\bigr)P(t). \]

This is the exact finite distinction between following a moving subspace and remaining in a fixed one.

Density matrices require separate packet labels for their output and input coordinates. The superoperators

\[ {\mathcal P}_{ab}(X)=P_aXP_b \]

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 \(U\otimes\overline U\).

Error correction is expressed through compression to a code. Let \(P\) be the code projector. The class of \(PAP\) modulo scalar multiples of \(P\) measures the obstruction to making \(A\) undetectable within the code. The Knill Laflamme criterion says that this obstruction vanishes for every product \(E_a^*E_b\) from the error family. Diagonalizing the resulting positive coefficient matrix produces orthogonal syndrome packets and an explicit recovery on the error span.

The effective dynamics section returns to ordinary Hilbert space packets. Schur complementation gives the exact retained resolvent and a frequency dependent self energy. The same hidden block gives a causal memory kernel in time. For a self adjoint hidden Hamiltonian, the self energy has a fixed sign in the upper half plane, and the time kernel is positive definite. Uniform spectral separation gives direct bounds that can be carried through a refinement tower.

The geometric part develops two finite constructions. A strictly positive contraction gives a reflection positive Hankel kernel, and its logarithm reconstructs a nonnegative Hamiltonian. A finite Laplacian gives a heat trace from which the spectral dimension is defined at each diffusion scale. The article proves convergence from periodic lattices to the continuum torus at every fixed positive diffusion time. Poisson summation then determines the continuum dimension as diffusion time tends to zero.

Constraint closure needs a different obstruction. If \(C_a\) are self adjoint constraints and \(P_{{\mathrm{phys}}}\) projects onto their common kernel, then

\[ P_{{\mathrm{phys}}}[C_a,C_b]P_{{\mathrm{phys}}}=0 \]

holds automatically. This compression cannot test closure. The correct finite obstruction is the class of \([C_a,C_b]\) in the quotient of \({\operatorname{End}}({\mathcal H})\) by the span of the constraints. Vanishing means that the constraint span is closed under commutators.

The last part gives two theorems for towers of bounded Hamiltonians. Exactly compatible finite Hamiltonians with a uniform norm bound define a bounded self adjoint operator on the Hilbert direct limit. When successive bounded self adjoint Hamiltonians on a common Hilbert space have norm summable differences, they converge to a bounded self adjoint limit. Their unitary evolutions, and their Euclidean evolutions under the stated positivity assumption, converge uniformly on bounded time intervals. Reflection positivity and closure with fixed structure coefficients pass to the limit under the stated convergence assumptions.

These results do not construct an interacting continuum quantum field theory. They do not prove anomaly freedom on an unbounded common domain, Osterwalder Schrader reconstruction for a continuum measure, an Einstein infrared action, or a positive massless spin two sector. Each of those conclusions needs hypotheses absent from the finite calculations.

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A sequence of finite quantum layers leading to separate continuum-limit criteria. — The finite mathematical layers developed in the article and the separate limit criteria needed at the continuum boundary.

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The packet coordinates used in the article.
Object	Packet coordinate	Exact structural condition
State	\(P_a\psi\)	completeness of the projector family
Operator	\(P_bAP_a\)	block multiplication
Circuit	intermediate packet path	insertion of complete resolutions
Density matrix	\(P_a\rho P_b\)	two sided operator packets
Channel	superoperator block	complete positivity and covariance when declared
Code	syndrome packet	scalar compression of error products
Effective model	Feshbach map	invertibility of the hidden resolvent
Euclidean model	positive transfer matrix	positivity and self adjointness
Constraint system	quotient obstruction	commutator closure modulo the constraint span
Continuum tower	direct or norm limit	compatibility plus explicit convergence

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II.2 Quantum seeds and observable algebras

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Definition II.2.1: Finite quantum seed system

A finite quantum seed system is a triple

\[ ({\mathcal H},{\mathcal A},S), \]

where \({\mathcal H}\) is a finite dimensional Hilbert space, \({\mathcal A}\subseteq{\operatorname{End}}({\mathcal H})\) is a unital star subalgebra, and \(S\subseteq{\mathcal H}\) is a finite set. Its generated hull is

\[ {\mathcal A} S = \operatorname{span}\{A\psi:A\in{\mathcal A},\ \psi\in S\}. \]

The seed set is cyclic when \({\mathcal A} S={\mathcal H}\). The quantum seed rank is the least cardinality of a cyclic seed set.

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The algebra is part of the data. A vector may be cyclic for \({\operatorname{End}}({\mathcal H})\) and fail to be cyclic for a restricted observable or gate algebra.

Let

\[ {\mathcal A} \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}({\mathbb C})\otimes I_{m_r}\bigr) \]

act on

\[ {\mathcal H} \cong \bigoplus_{r=1}^{s} \bigl({\mathbb C}^{d_r}\otimes{\mathbb C}^{m_r}\bigr). \]

Assume \(d_r,m_r\ge1\) 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.

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Theorem II.2.2: Orbit dimension of one seed

For integers \(d\ge1\), \(m\ge0\), and the action of \(M_d({\mathbb C})\otimes I_m\) on \({\mathbb C}^d\otimes{\mathbb C}^m\), let \(C_\psi\) be the coefficient matrix of \(\psi\). Then

\[ \dim\operatorname{span}\{(A\otimes I_m)\psi:A\in M_d({\mathbb C})\} = d{\operatorname{rank}} C_\psi. \]

The vector is cyclic exactly when \({\operatorname{rank}} C_\psi=m\).

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Proof of Theorem II.2.2

Under the coefficient matrix identification, \((A\otimes I_m)\psi\) corresponds to \(AC_\psi\). Every row of \(AC_\psi\) lies in the row space of \(C_\psi\). Conversely, a matrix whose rows lie in that row space equals \(AC_\psi\) for a suitable \(A\). There are \(d\) independent row choices in a space of dimension \({\operatorname{rank}} C_\psi\). The orbit dimension is therefore \(d{\operatorname{rank}} C_\psi\). It equals \(dm\) exactly when \(C_\psi\) has full column rank.

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Theorem II.2.3: Quantum seed rank

For the displayed multiplicity decomposition,

\[ \operatorname{srank}_{{\mathcal A}}({\mathcal H}) = \max_{1\le r\le s} \left\lceil\frac{m_r}{d_r}\right\rceil. \]

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Proof of Theorem II.2.3

A single seed contributes a row space of dimension at most \(d_r\) to the \(r\)th multiplicity factor. Hence \(k\) seeds cover at most \(kd_r\) multiplicity directions, which gives the lower bound.

Set

\[ k=\max_r\left\lceil\frac{m_r}{d_r}\right\rceil. \]

For each \(r\), partition a basis of \({\mathbb C}^{m_r}\) into at most \(k\) groups of size at most \(d_r\). Choose the coefficient matrix of the \(j\)th seed so that its row space spans the \(j\)th group. The combined row spaces span every multiplicity factor. The preceding theorem then shows that the \(k\) seeds generate every summand.

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Corollary II.2.4: Local orbit and Schmidt rank

Let \(\psi\in{\mathcal H}_A\otimes{\mathcal H}_B\), let \(d_A=\dim{\mathcal H}_A\ge1\), and let \(r\) be the Schmidt rank of \(\psi\). Then

\[ \dim\operatorname{span}\{(A\otimes I_B)\psi:A\in{\operatorname{End}}({\mathcal H}_A)\} = d_A r. \]

Local operators on \(A\) generate the whole bipartite Hilbert space exactly when the reduced state on \(B\) has full support.

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Proof of Corollary II.2.4

The Schmidt rank is the rank of the coefficient matrix in any product basis. Substituting that rank into Theorem II.2.2 gives the stated orbit dimension.

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Local orbit dimension increases linearly with Schmidt rank. — The dimension reached by all local operators is linear in Schmidt rank.

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Proposition II.2.5: Computational address orbit

For \(n\) qubits, let

\[ \psi_0={\left|0\right\rangle}^{\otimes n} \]

and let \(X_j\) be the Pauli \(X\) operator on the \(j\)th qubit. Then

\[ \left\{ X_1^{a_1}\cdots X_n^{a_n}\psi_0: (a_1,\ldots,a_n)\in\{0,1\}^n \right\} \]

is the computational basis.

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Proof of Proposition II.2.5

The addressed vector is \({\left|a_1\cdots a_n\right\rangle}\). Distinct binary addresses produce the \(2^n\) distinct basis vectors.

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Remark II.2.6

The seed rank records generation by a declared algebra. It does not measure circuit depth, locality cost, control complexity, or physical reachability under a restricted time dependent Hamiltonian.

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II.3 Symmetry packets and selection rules

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Let \(G\) be a finite group and let

\[ U:G\longrightarrow{\mathcal U}({\mathcal H}) \]

be a unitary representation. Write \(\widehat G\) for the set of irreducible unitary representations. For \(\lambda\in\widehat G\), let \(d_\lambda\) and \(\chi_\lambda\) denote its dimension and character.

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Theorem II.3.1: Finite group isotypic projectors

The operators

\[ P_\lambda = \frac{d_\lambda}{{\left\lvert G\right\rvert}} \sum_{g\in G} \overline{\chi_\lambda(g)}U(g) \]

are orthogonal projectors. They satisfy

\[ P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda, \qquad \sum_{\lambda\in\widehat G}P_\lambda=I_{{\mathcal H}}. \]

The range of \(P_\lambda\) is the \(\lambda\) isotypic component of \({\mathcal H}\).

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Proof of Theorem II.3.1

Complete reducibility gives a unitary decomposition

\[ {\mathcal H} \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}. \]

On the \(\mu\) summand, the displayed average is central in the irreducible representation \(\pi_\mu\), so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when \(\mu=\lambda\) and zero otherwise. The asserted identities follow on the direct sum.

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The central projectors separate irreducible types but do not choose coordinates inside the representation factor. Matrix units provide that finer resolution.

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Theorem II.3.2: Representation matrix units

Fix an irreducible unitary representation \(\pi_\lambda:G\to U(d_\lambda)\) and an orthonormal basis of its representation space. Define

\[ E_{ij}^{\lambda} = \frac{d_\lambda}{{\left\lvert G\right\rvert}} \sum_{g\in G} \bigl[\pi_\lambda(g^{-1})\bigr]_{ji}U(g). \]

Then

\[ E_{ij}^{\lambda}E_{k\ell}^{\mu} = \delta_{\lambda\mu}\delta_{jk}E_{i\ell}^{\lambda}, \qquad \bigl(E_{ij}^{\lambda}\bigr)^*=E_{ji}^{\lambda}, \]

and

\[ P_\lambda=\sum_{i=1}^{d_\lambda}E_{ii}^{\lambda}. \]

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Proof of Theorem II.3.2

Apply the matrix coefficient form of Schur orthogonality to the two group averages. It gives the matrix unit multiplication law. Unitarity gives the adjoint relation after replacing \(g\) by \(g^{-1}\). Summing the diagonal coefficients gives the character average in Theorem II.3.1.

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II.3.1 Cyclic charge packets

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Let \(N\ge1\) be an integer, \(S\) unitary with \(S^N=I\), and \(\omega={\mathrm e}^{2\pi{\mathrm i}/N}\).

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Corollary II.3.3: Cyclic charge projectors

The operators

\[ P_j = \frac1N \sum_{r=0}^{N-1} \omega^{-jr}S^r, \qquad j\in{\mathbb Z}/N{\mathbb Z}, \]

form a complete orthogonal packet system and satisfy

\[ SP_j=\omega^jP_j. \]

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Proof of Corollary II.3.3

Apply Theorem II.3.1 to the cyclic group. Its irreducible representations are the one dimensional characters \(r\mapsto\omega^{jr}\).

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Definition II.3.4: Operator charge

An operator \(A\in{\operatorname{End}}({\mathcal H})\) has cyclic charge \(q\in{\mathbb Z}/N{\mathbb Z}\) when

\[ SAS^*=\omega^qA. \]

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Theorem II.3.5: Charge selection rule

If \(A\) has charge \(q\), then

\[ P_kAP_j=0 \]

unless \(k=j+q\) in \({\mathbb Z}/N{\mathbb Z}\). In fact,

\[ AP_j=P_{j+q}AP_j. \]

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Proof of Theorem II.3.5

For \(x\in P_j{\mathcal H}\),

\[ S(Ax) =(SAS^*)Sx =\omega^{q+j}Ax. \]

Thus \(Ax\in P_{j+q}{\mathcal H}\). The block statement follows.

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Corollary II.3.6: Charge addition

If \(A\) and \(B\) have charges \(q\) and \(r\), then \(AB\) has charge \(q+r\). The adjoint \(A^*\) has charge \(-q\).

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Proof of Corollary II.3.6

Conjugation by \(S\) is a star automorphism. Applying it to \(AB\) adds the two charges, while applying it to \(A^*\) changes the sign of the charge.

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A charge-one operator advances through four cyclic symmetry packets. — An operator of charge one moves each cyclic packet to the next packet and has no other transfer blocks.

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II.3.2 Tensor refinement

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Suppose the \(a\)th tensor factor carries a cyclic packet family

\[ \bigl(P_j^{(a)}\bigr)_{j\in{\mathbb Z}/N{\mathbb Z}}. \]

The fine tensor packets are

\[ P_{{\boldsymbol{j}}}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}. \]

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Theorem II.3.7: Tensor charge refinement

For \(r\in{\mathbb Z}/N{\mathbb Z}\), define

\[ P_r^{\rm total} = \sum_{j_1+\cdots+j_m=r} P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}. \]

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.

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Proof of Theorem II.3.7

The fine tensor projectors are mutually orthogonal and sum to the identity. The total residue classes partition the fine address set. Summing the fine projectors over each class gives the result.

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Corollary II.3.8: Tensor selection rule

If \(A_a\) has local charge \(q_a\), then

\[ A_1\otimes\cdots\otimes A_m \]

has total charge \(q_1+\cdots+q_m\).

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Proof of Corollary II.3.8

Conjugate the tensor product by the total cyclic symmetry and multiply the local phase factors.

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Remark II.3.9

For a nonabelian group, tensor products are refined by irreducible fusion multiplicities rather than residue addition. The central isotypic projectors remain exact, but a complete tensor refinement also requires a choice of Clebsch Gordan intertwiners.

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II.4 Quantum circuits as packet paths

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Let

\[ U=U_mU_{m-1}\cdots U_1 \]

be a finite circuit with \(m\ge1\), and let \((P_a)_{a\in I}\) be a complete orthogonal packet family. For the empty circuit, \(U=I\) and its transfer block is \(P_bP_a=\delta_{ba}P_a\), whose endpoint sum reconstructs \(I\).

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Theorem II.4.1: Exact circuit path expansion

For source packet \(a_0\) and target packet \(a_m\),

\[ P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}. \]

Summing over \(a_0\) and \(a_m\) reconstructs \(U\).

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Proof of Theorem II.4.1

Insert \(I=\sum_aP_a\) between every adjacent pair of gates and distribute the finite sums. Completeness at the two endpoints gives the full operator.

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For a packet path

\[ \gamma=(a_0,a_1,\ldots,a_m), \]

write

\[ U_\gamma = P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}. \]

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Theorem II.4.2: Packet unitarity identities

If \(U\) is unitary and \(U_{ba}=P_bUP_a\), then

\[ \sum_bU_{ba}^*U_{bc}=\delta_{ac}P_a \]

and

\[ \sum_aU_{ba}U_{da}^*=\delta_{bd}P_b. \]

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Proof of Theorem II.4.2

Use completeness and unitarity:

\[ \sum_bU_{ba}^*U_{bc} = P_aU^*\left(\sum_bP_b\right)UP_c =P_aP_c. \]

Summing instead over the source packets and using unitarity in the reverse order gives the second identity.

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Corollary II.4.3: Probability conservation from one packet

If \(\psi\in P_a{\mathcal H}\), then

\[ \sum_b{\left\lVert P_bU\psi\right\rVert}^2={\left\lVert\psi\right\rVert}^2. \]

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Proof of Corollary II.4.3

Take the inner product of the first identity in Theorem II.4.2 with \(\psi\) and set \(c=a\).

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Theorem II.4.4: Path interference formula

Fix a source vector \(\psi\in P_{a_0}{\mathcal H}\) and a target packet \(b\). Let \(\Gamma_b\) be the set of packet paths from \(a_0\) to \(b\), and set

\[ \eta_\gamma=U_\gamma\psi. \]

Choose any total order on this finite path set. Then

\[ {\left\lVert P_bU\psi\right\rVert}^2 = \sum_{\gamma\in\Gamma_b}{\left\lVert\eta_\gamma\right\rVert}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} {\left\langle\eta_\gamma,\eta_{\gamma'}\right\rangle}. \]

Consequently,

\[ {\left\lVert P_bU\psi\right\rVert} \leq \sum_{\gamma\in\Gamma_b}{\left\lVert\eta_\gamma\right\rVert}. \]

If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.

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Proof of Theorem II.4.4

The path expansion gives

\[ P_bU\psi=\sum_{\gamma\in\Gamma_b}\eta_\gamma. \]

Expanding the squared norm and grouping conjugate cross terms gives the interference formula. The triangle inequality gives the norm bound. When the path outputs are orthogonal, every cross term vanishes and only the individual path probabilities remain.

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Proposition II.4.5: Charge pruning of circuit paths

Suppose every gate has a finite charge decomposition

\[ U_r=\sum_qU_r^{(q)}, \qquad SU_r^{(q)}S^*=\omega^qU_r^{(q)}. \]

A term with gate charges \(q_1,\ldots,q_m\) can contribute from source charge \(a_0\) only to target charge

\[ a_m=a_0+q_1+\cdots+q_m. \]

Every packet path violating one of the intermediate charge updates vanishes.

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Proof of Proposition II.4.5

Apply Theorem II.3.5 after each gate component. The packet address is forced at every step.

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Allowed packet paths across a four-gate circuit. — A circuit block is the coherent sum of its allowed packet paths. Selection rules remove forbidden paths before amplitudes are evaluated.

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Example II.4.6: Bell circuit

Starting from \({\left|00\right\rangle}\), apply a Hadamard gate to the first qubit and then a controlled NOT gate. The final state is

\[ \frac{{\left|00\right\rangle}+{\left|11\right\rangle}}{\sqrt2}. \]

It lies in the even parity packet. The intermediate state has one even and one odd computational contribution, so the final even amplitude is reconstructed by more than one packet path.

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II.5 Symmetry resolved dynamics and leakage

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Let \(H=H^*\) and let \((P_a)_{a\in I}\) be a complete orthogonal packet system.

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Theorem II.5.1: Exact invariant packets

The following conditions are equivalent:

The Hamiltonian commutes with each projector: \([H,P_a]=0\) for every \(a\).
Transfer blocks between distinct packets vanish: \(P_bHP_a=0\) whenever \(a\neq b\).
Every packet \(P_a{\mathcal H}\) is invariant under \({\mathrm e}^{-{\mathrm i} tH}\) for every real \(t\).

When they hold,

\[ H=\sum_aP_aHP_a \]

and

\[ {\mathrm e}^{-{\mathrm i} tH}=\sum_aP_a{\mathrm e}^{-{\mathrm i} tH}P_a. \]

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Proof of Theorem II.5.1

Commutation is equivalent to vanishing of every off diagonal block. The same block decomposition passes to polynomials in \(H\), then to the exponential by norm convergence. Invariance of every packet implies invariance under both \({\mathrm e}^{-{\mathrm i} tH}\) and its adjoint. Differentiation at \(t=0\) gives commutation with \(H\).

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Fix one packet projector \(P\) and set \(Q=I-P\).

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Lemma II.5.2: Off diagonal norm

For self adjoint \(H\),

\[ {\left\lVert[H,P]\right\rVert}={\left\lVert QHP\right\rVert}={\left\lVert PHQ\right\rVert}. \]

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Proof of Lemma II.5.2

Relative to \(P{\mathcal H}\oplus Q{\mathcal H}\),

\[ [H,P] = \begin{pmatrix} 0&-PHQ\\ QHP&0 \end{pmatrix}. \]

Its square has diagonal blocks \(-PHQHP\) and \(-QHPHQ\). The norm is therefore the largest singular value of \(QHP\).

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Theorem II.5.3: Fixed packet leakage

For every real \(t\),

\[ {\left\lVert Q{\mathrm e}^{-{\mathrm i} tH}P\right\rVert} \leq {\left\lvert t\right\rvert}{\left\lVert QHP\right\rVert} = {\left\lvert t\right\rvert}{\left\lVert[H,P]\right\rVert}. \]

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Proof of Theorem II.5.3

Let

\[ H_{\rm d}=PHP+QHQ, \qquad H_{\rm o}=PHQ+QHP. \]

The propagator generated by \(H_{\rm d}\) preserves \(P{\mathcal H}\) and \(Q{\mathcal H}\). Duhamel's formula relative to \(H_{\rm d}\) gives

\[ Q{\mathrm e}^{-{\mathrm i} tH}P = -{\mathrm i} \int_0^t Q{\mathrm e}^{-{\mathrm i}(t-s)H}H_{\rm o}{\mathrm e}^{-{\mathrm i} sH_{\rm d}}P {\,\mathrm d} s. \]

Since \(H_{\rm o}P=QHP\) and both propagators are unitary, the norm is at most \({\left\lvert t\right\rvert}{\left\lVert QHP\right\rVert}\). Use Lemma II.5.2 for the last equality.

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An exact two-packet transition amplitude lies below a commutator envelope. — Exact transition amplitude in a two packet model and the linear commutator bound.

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Corollary II.5.4: Leakage probability

If \(\psi\in P{\mathcal H}\) is a unit vector, then

\[ {\left\lVert Q{\mathrm e}^{-{\mathrm i} tH}\psi\right\rVert}^2 \leq t^2{\left\lVert QHP\right\rVert}^2. \]

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Proof of Corollary II.5.4

Apply Theorem II.5.3 to the unit vector, then square the resulting nonnegative bound.

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II.6 Moving packets and geometric transport

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Let \((P_a(t))_{a\in I}\) be a twice continuously differentiable family of complete orthogonal packet systems on a finite dimensional Hilbert space.

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Theorem II.6.1: Canonical packet connection

Define

\[ K(t) = \sum_a\dot P_a(t)P_a(t). \]

Then

\[ K(t)^*=-K(t) \]

and

\[ [K(t),P_a(t)]=\dot P_a(t) \]

for every \(a\).

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Proof of Theorem II.6.1

Differentiating \(P_a^2=P_a\) gives

\[ \dot P_aP_a+P_a\dot P_a=\dot P_a. \]

Summing over \(a\) and using \(\sum_a\dot P_a=0\) yields

\[ \sum_aP_a\dot P_a=-\sum_a\dot P_aP_a. \]

This proves skew adjointness. For fixed \(b\),

\[ KP_b=\dot P_bP_b. \]

Differentiating \(P_bP_a=0\) for \(a\neq b\) gives

\[ P_b\dot P_a=-\dot P_bP_a. \]

Hence

\[ P_bK=-\dot P_b(I-P_b). \]

Subtracting gives \([K,P_b]=\dot P_b\).

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Theorem II.6.2: Exact Kato transport

Let \(W(t)\) solve

\[ \dot W(t)=K(t)W(t), \qquad W(0)=I. \]

Then \(W(t)\) is unitary and

\[ P_a(t)W(t)=W(t)P_a(0) \]

for every packet \(a\).

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Proof of Theorem II.6.2

Skew adjointness gives

\[ \frac{{\,\mathrm d}}{{\,\mathrm d} t}W(t)^*W(t)=0. \]

Thus \(W(t)\) is unitary. Next,

\[ \frac{{\,\mathrm d}}{{\,\mathrm d} t} \left(W(t)^*P_a(t)W(t)\right) = W(t)^* \left( \dot P_a-[K,P_a] \right) W(t)=0. \]

The expression equals \(P_a(0)\) at \(t=0\), which proves the intertwining identity.

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Theorem II.6.3: Moving frame Hamiltonian

Let \(U(t)\) solve

\[ {\mathrm i}\dot U(t)=H(t)U(t), \qquad U(0)=I, \]

where \(H(t)=H(t)^*\) is continuous on the compact time interval under consideration. Set

\[ V(t)=W(t)^*U(t). \]

Then

\[ {\mathrm i}\dot V(t)=H_{\rm mov}(t)V(t), \]

where

\[ H_{\rm mov}(t) = W(t)^*H(t)W(t)-{\mathrm i} W(t)^*\dot W(t) = W(t)^*\bigl(H(t)-{\mathrm i} K(t)\bigr)W(t). \]

The moving frame Hamiltonian is self adjoint.

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Proof of Theorem II.6.3

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 \(-{\mathrm i} W^*\dot W\) is self adjoint because \(W^*\dot W\) is skew adjoint.

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Theorem II.6.4: Moving packet leakage

Fix a packet \(P(t)=P_a(t)\) and put \(Q(t)=I-P(t)\). For \(0\leq t\leq T\),

\[ {\left\lVert Q(t)U(t)P(0)\right\rVert} \leq \int_0^t {\left\lVert Q(s)\bigl(H(s)-{\mathrm i} K(s)\bigr)P(s)\right\rVert} {\,\mathrm d} s. \]

In particular, the physical evolution follows the moving packet exactly on \([0,T]\) when the displayed off packet block vanishes for every time in that interval.

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Proof of Theorem II.6.4

In the Kato frame, \(P(0)\) is fixed and

\[ Q(t)U(t)P(0) = W(t)Q(0)V(t)P(0). \]

Apply the time dependent version of the Duhamel argument in Theorem II.5.3 to \(H_{\rm mov}(t)\). The off diagonal block is

\[ Q(0)H_{\rm mov}(s)P(0) = W(s)^*Q(s)\bigl(H(s)-{\mathrm i} K(s)\bigr)P(s)W(s). \]

Unitary conjugation preserves the norm.

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The exact coupling test therefore includes the geometric term. Even if the Hamiltonian is diagonal in the packet decomposition at every fixed time, its evolution can fail to follow a packet whose range rotates.

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II.6.1 Discrete packet transport

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Let \((P_a)\) and \((Q_a)\) be two complete packet systems with the same index set. Define

\[ S({\mathcal Q},{\mathcal P})=\sum_aQ_aP_a. \]

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Theorem II.6.5: Direct packet rotation

Assume

\[ {\left\lVert S({\mathcal Q},{\mathcal P})-I\right\rVert}<1. \]

Then \(S({\mathcal Q},{\mathcal P})\) is invertible. Its unitary polar factor

\[ {\mathcal U}({\mathcal Q},{\mathcal P}) = S(S^*S)^{-1/2} \]

satisfies

\[ Q_a{\mathcal U}({\mathcal Q},{\mathcal P}) = {\mathcal U}({\mathcal Q},{\mathcal P})P_a \]

for every \(a\).

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Proof of Theorem II.6.5

The norm condition gives invertibility by the Neumann series. The identity

\[ Q_aS=SP_a=Q_aP_a \]

shows that \(S^*S\) commutes with every \(P_a\). Its inverse square root also commutes with every \(P_a\). Hence

\[ Q_a{\mathcal U} =Q_aS(S^*S)^{-1/2} =SP_a(S^*S)^{-1/2} ={\mathcal U} P_a. \]

The polar factor is unitary because \(S\) is invertible.

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Theorem II.6.6: Convergence of short packet rotations

Let \((P_a(t))\) be twice continuously differentiable on \([0,T]\). For a partition

\[ 0=t_0<t_1<\cdots<t_n=T, \]

let

\[ W_\pi = {\mathcal U}\bigl({\mathcal P}(t_n),{\mathcal P}(t_{n-1})\bigr) \cdots {\mathcal U}\bigl({\mathcal P}(t_1),{\mathcal P}(t_0)\bigr). \]

For sufficiently fine partitions the factors are defined, and

\[ {\left\lVert W_\pi-W(T)\right\rVert} \leq C_T{\left\lvert\pi\right\rvert} \]

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.

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Proof of Theorem II.6.6

Taylor expansion gives

\[ S\bigl({\mathcal P}(t+h),{\mathcal P}(t)\bigr) = I+hK(t)+O(h^2) \]

uniformly on \([0,T]\). Since \(K(t)^*=-K(t)\),

\[ S^*S=I+O(h^2), \]

so the polar factor satisfies

\[ {\mathcal U}\bigl({\mathcal P}(t+h),{\mathcal P}(t)\bigr) = I+hK(t)+O(h^2). \]

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 \(\sum_j(t_{j+1}-t_j)^2\), which is at most \(T{\left\lvert\pi\right\rvert}\).

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A moving rank-one packet traced on the Bloch sphere. — A moving rank one packet on the Bloch sphere. The Kato evolution transports the range without choosing a phase convention for its eigenvector.

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II.6.2 Local frames and curvature

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Let \(P(x)\) be a smooth rank \(r\) orthogonal projector over a parameter manifold. Choose a local isometry

\[ V(x):{\mathbb C}^r\longrightarrow{\mathcal H}, \qquad V(x)V(x)^*=P(x). \]

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Theorem II.6.7: Nonabelian packet connection

The one form with matrix values

\[ {\mathcal A}=V^*{\,\mathrm d} V \]

is skew adjoint. Under a frame change \(V'=VG\), where \(G(x)\in U(r)\),

\[ {\mathcal A}' = G^*{\mathcal A} G+G^*{\,\mathrm d} G. \]

Its curvature

\[ {\mathcal F}={\,\mathrm d}{\mathcal A}+{\mathcal A}\wedge{\mathcal A} \]

satisfies

\[ {\mathcal F}=V^*({\,\mathrm d} P\wedge{\,\mathrm d} P)V. \]

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Proof of Theorem II.6.7

Differentiate \(V^*V=I\) to obtain \({\mathcal A}^*=-{\mathcal A}\). The gauge transformation law follows by differentiating \(VG\). Since \(P=VV^*\) and \((I-P){\,\mathrm d} V={\,\mathrm d} P\,V\), direct expansion gives

\[ {\,\mathrm d}{\mathcal A}+{\mathcal A}\wedge{\mathcal A} = {\,\mathrm d} V^*(I-P)\wedge{\,\mathrm d} V = V^*({\,\mathrm d} P\wedge{\,\mathrm d} P)V. \]

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Corollary II.6.8: Bloch sphere curvature

Let

\[ P(\theta,\phi) = \frac12\left(I+{\boldsymbol{n}}(\theta,\phi)\cdot{\boldsymbol{\sigma}}\right), \]

where

\[ {\boldsymbol{n}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta). \]

With the convention

\[ {\mathcal F}_{\theta\phi} =-{\mathrm i}{\operatorname{Tr}}\bigl(P[\partial_\theta P,\partial_\phi P]\bigr), \]

one has

\[ {\mathcal F}_{\theta\phi}=\frac12\sin\theta \]

and

\[ \frac1{2\pi} \int_0^{2\pi}\int_0^\pi {\mathcal F}_{\theta\phi}{\,\mathrm d}\theta{\,\mathrm d}\phi =1. \]

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Proof of Corollary II.6.8

The Pauli identity

\[ ({\boldsymbol{a}}\cdot{\boldsymbol{\sigma}})({\boldsymbol{b}}\cdot{\boldsymbol{\sigma}}) =({\boldsymbol{a}}\cdot{\boldsymbol{b}})I +{\mathrm i}({\boldsymbol{a}}\times{\boldsymbol{b}})\cdot{\boldsymbol{\sigma}} \]

gives

\[ -{\mathrm i}{\operatorname{Tr}}\bigl(P[\partial_\theta P,\partial_\phi P]\bigr) =\frac12 {\boldsymbol{n}}\cdot (\partial_\theta{\boldsymbol{n}}\times\partial_\phi{\boldsymbol{n}}) =\frac12\sin\theta. \]

Integrating the displayed curvature over the sphere and applying the stated normalization gives one.

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II.7 Operator packets, measurements, and quantum channels

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Density matrices and observables carry both an input packet and an output packet. Equip \({\operatorname{End}}({\mathcal H})\) with the Hilbert Schmidt inner product

\[ {\left\langle X,Y\right\rangle}_{{\mathrm{HS}}}={\operatorname{Tr}}(X^*Y). \]

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Definition II.7.1: Two sided operator packet

For a complete orthogonal family \((P_a)\), define

\[ {\mathcal P}_{ab}(X)=P_aXP_b. \]

The range

\[ {\mathcal B}_{ab}=P_a{\operatorname{End}}({\mathcal H})P_b \]

is the operator packet with output address \(a\) and input address \(b\).

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Theorem II.7.2: Complete operator packet system

The superoperators \(({\mathcal P}_{ab})_{a,b\in I}\) are mutually orthogonal projections on Hilbert Schmidt operator space. They satisfy

\[ X=\sum_{a,b}P_aXP_b \]

and

\[ {\left\lVert X\right\rVert}_{{\mathrm{HS}}}^2 = \sum_{a,b}{\left\lVert P_aXP_b\right\rVert}_{{\mathrm{HS}}}^2. \]

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Proof of Theorem II.7.2

Idempotence and completeness follow from the projector identities. For Hilbert Schmidt orthogonality,

\[ \begin{aligned} {\left\langle P_aXP_b,P_cYP_d\right\rangle}_{{\mathrm{HS}}} &= {\operatorname{Tr}}(P_bX^*P_aP_cYP_d)\\ &= \delta_{ac}\delta_{bd} {\operatorname{Tr}}(P_bX^*P_aYP_b). \end{aligned} \]

Since the packets are orthogonal and sum to the original operator, their squared norms add to its squared norm.

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An operator matrix is divided into diagonal and off-diagonal packet blocks. — A density matrix or observable carries two packet addresses. Diagonal packets contain populations and internal observables. Off diagonal packets contain coherences.

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Definition II.7.3: Packet dephasing

The packet dephasing map is

\[ {\mathcal D}_{{\mathcal P}}(X)=\sum_aP_aXP_a. \]

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Theorem II.7.4: Dephasing as conditional expectation

The map \({\mathcal D}_{{\mathcal P}}\) is completely positive, trace preserving, unital, and idempotent. It is the Hilbert Schmidt orthogonal projection onto the block diagonal algebra

\[ {\mathcal A}_{{\mathcal P}} = \{X:[X,P_a]=0\text{ for every }a\}. \]

In particular,

\[ {\left\lVert{\mathcal D}_{{\mathcal P}}(X)\right\rVert}_{{\mathrm{HS}}} \leq {\left\lVert X\right\rVert}_{{\mathrm{HS}}}. \]

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Proof of Theorem II.7.4

The projectors \(P_a\) form a Kraus family because \(\sum_aP_a^*P_a=I\). Thus the map is completely positive and trace preserving. It is unital for the same reason. Orthogonality and idempotence follow from Theorem II.7.2. Its range is the sum of the diagonal operator packets. This is exactly the commutant of the projector family.

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Corollary II.7.5: Block diagonal measurement invariance

Let \((E_k)\) be a positive operator valued measurement, and suppose

\[ E_k={\mathcal D}_{{\mathcal P}}(E_k) \]

for every \(k\). Then every density matrix \(\rho\) satisfies

\[ {\operatorname{Tr}}(E_k\rho) = {\operatorname{Tr}}\bigl(E_k{\mathcal D}_{{\mathcal P}}(\rho)\bigr). \]

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Proof of Corollary II.7.5

Since \({\mathcal D}_{{\mathcal P}}\) is self adjoint for the Hilbert Schmidt inner product,

\[ {\operatorname{Tr}}(E_k\rho) = {\operatorname{Tr}}\bigl({\mathcal D}_{{\mathcal P}}(E_k)\rho\bigr) = {\operatorname{Tr}}\bigl(E_k{\mathcal D}_{{\mathcal P}}(\rho)\bigr). \]

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II.7.1 Channel superblocks

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Let \(\Phi:{\operatorname{End}}({\mathcal H}_X)\to{\operatorname{End}}({\mathcal H}_Y)\) be linear. Choose packet systems \((P_a)\) on \({\mathcal H}_X\) and \((Q_c)\) on \({\mathcal H}_Y\). Define

\[ \Phi_{cd,ab} = {\mathcal Q}_{cd}\Phi{\mathcal P}_{ab}. \]

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Theorem II.7.6: Exact channel block calculus

Every linear map has the exact reconstruction

\[ \Phi=\sum_{a,b,c,d}\Phi_{cd,ab}. \]

If \(\Psi:{\operatorname{End}}({\mathcal H}_Y)\to{\operatorname{End}}({\mathcal H}_Z)\), then

\[ (\Psi\Phi)_{ef,ab} = \sum_{c,d} \Psi_{ef,cd}\Phi_{cd,ab}. \]

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Proof of Theorem II.7.6

Insert the complete operator packet resolutions before and after \(\Phi\). For composition, insert the middle resolution between \(\Psi\) and \(\Phi\) and collect the output and input packets.

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Standard fact II.7.7: Finite Kraus and Choi criteria

A linear map \(\Phi:M_n({\mathbb C})\to M_m({\mathbb C})\) is completely positive exactly when it has a Kraus representation

\[ \Phi(X)=\sum_rK_rXK_r^*. \]

It is trace preserving exactly when

\[ \sum_rK_r^*K_r=I. \]

Equivalently, its Choi matrix

\[ J(\Phi) = \sum_{i,j}\Phi({\left|i\right\rangle}{\left\langle j\right|})\otimes{\left|i\right\rangle}{\left\langle j\right|} \]

is positive, and trace preservation is

\[ {\operatorname{Tr}}_{\rm out}J(\Phi)=I_{\rm in}. \]

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These finite criteria are classical [Stinespring1955], [Choi1975], [Kraus1983], [Watrous2018].

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II.7.2 Covariance and coherence order

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Let \(S\) be unitary and \(N\ge1\) an integer with \(S^N=I\), and define the conjugation action

\[ {\operatorname{Ad}}_S(X)=SXS^*. \]

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Theorem II.7.8: Coherence order projectors

For \(q\in{\mathbb Z}/N{\mathbb Z}\), define

\[ {\mathcal L}_q(X) = \frac1N \sum_{r=0}^{N-1} \omega^{-qr}S^rXS^{-r}. \]

Then the \({\mathcal L}_q\) form a complete orthogonal packet system on Hilbert Schmidt operator space. Their ranges are the operators satisfying

\[ SXS^*=\omega^qX. \]

If \(P_j\) are the state charge projectors, then

\[ {\operatorname{Ran}}{\mathcal L}_q = \bigoplus_jP_{j+q}{\operatorname{End}}({\mathcal H})P_j. \]

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Proof of Theorem II.7.8

Apply the cyclic character projector formula to the unitary representation \({\operatorname{Ad}}_S\) on operator space. The last identity follows from Theorem II.3.5 applied to each operator packet.

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Theorem II.7.9: Covariant channels preserve coherence order

Suppose

\[ \Phi(SXS^*)=S\Phi(X)S^* \]

for every \(X\). Then

\[ \Phi{\mathcal L}_q={\mathcal L}_q\Phi \]

for every \(q\). Thus \(\Phi\) cannot mix distinct coherence orders.

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Proof of Theorem II.7.9

The covariance identity says that \(\Phi\) commutes with every power of \({\operatorname{Ad}}_S\). It therefore commutes with each finite character average \({\mathcal L}_q\).

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Theorem II.7.10: Choi symmetry criterion

Let \(U:G\to{\mathcal U}({\mathcal H})\) be a finite group representation. A channel \(\Phi\) is covariant,

\[ \Phi\bigl(U(g)XU(g)^*\bigr) = U(g)\Phi(X)U(g)^*, \]

if and only if

\[ \bigl(U(g)\otimes\overline{U(g)}\bigr) J(\Phi) \bigl(U(g)^*\otimes U(g)^{\mathsf T}\bigr) = J(\Phi) \]

for every \(g\in G\).

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Proof of Theorem II.7.10

Let

\[ \Omega=\sum_i{\left|i\right\rangle}\otimes{\left|i\right\rangle}. \]

Then

\[ J(\Phi)=(\Phi\otimes{\mathrm{id}})({\left|\Omega\right\rangle}{\left\langle\Omega\right|}). \]

The vector \(\Omega\) is invariant under \(U\otimes\overline U\). Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of

\[ {\operatorname{Ad}}_{U(g)}\circ\Phi\circ{\operatorname{Ad}}_{U(g)^*}. \]

The Choi correspondence is injective, so invariance implies that this map equals \(\Phi\), which is covariance.

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Corollary II.7.11: Choi packet decomposition

The Choi matrix of a covariant channel is block diagonal with respect to the isotypic decomposition of \(U\otimes\overline U\).

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Proof of Corollary II.7.11

Apply Theorem II.3.1 to the representation \(U\otimes\overline U\) and use Theorem II.7.10.

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Example II.7.12: Amplitude damping

For \(0\leq\gamma\leq1\), let

\[ K_0= \begin{pmatrix} 1&0\\ 0&\sqrt{1-\gamma} \end{pmatrix}, \qquad K_1= \begin{pmatrix} 0&\sqrt\gamma\\ 0&0 \end{pmatrix}. \]

The channel

\[ \Phi_\gamma(\rho)=K_0\rho K_0^*+K_1\rho K_1^* \]

is trace preserving. It sends the off diagonal packet \({\left|0\right\rangle}{\left\langle1\right|}\) to \(\sqrt{1-\gamma}{\left|0\right\rangle}{\left\langle1\right|}\), while the excited population transfers partly into the ground population.

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Amplitude damping changes populations and reduces the coherence magnitude. — Amplitude damping transfers population and contracts the coherence packet.

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II.7.3 Quantum dynamical semigroups

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Standard fact II.7.13: Finite Lindblad generator

A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schrödinger picture, has a generator of the form

\[ {\mathcal G}(X) = -{\mathrm i}[H,X] + \sum_r \left( L_rXL_r^* - \frac12L_r^*L_rX - \frac12XL_r^*L_r \right), \]

where \(H=H^*\). Conversely, every operator of this form generates a completely positive trace preserving semigroup [Lindblad1976].

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Theorem II.7.14: Charge preservation by a Lindblad generator

Let \(S\) be unitary and \(N\ge1\) an integer with \(S^N=I\). Assume

\[ SHS^*=H \]

and, for every jump operator, there is a charge \(q_r\) such that

\[ SL_rS^*=\omega^{q_r}L_r. \]

Then the Lindblad generator \({\mathcal G}\) preserves every coherence order:

\[ {\mathcal G}{\mathcal L}_p={\mathcal L}_p{\mathcal G}. \]

The quantum dynamical semigroup \({\mathrm e}^{t{\mathcal G}}\) has the same property.

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Proof of Theorem II.7.14

Let \(X\) have coherence order \(p\). The Hamiltonian commutator has order \(p\) because \(H\) has charge zero. The jump term \(L_rXL_r^*\) has charge

\[ q_r+p-q_r=p. \]

Also \(L_r^*L_r\) has charge zero, so both anticommutator terms have charge \(p\). Thus \({\mathcal G}\) preserves \({\operatorname{Ran}}{\mathcal L}_p\). The exponential series then shows that \({\mathrm e}^{t{\mathcal G}}\) preserves the same packets.

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Corollary II.7.15: Independent coherence evolution

Under the hypotheses of Theorem II.7.14, each coherence component of an initial operator evolves within its own order. In particular, a density matrix in the zero order sector remains in that sector.

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Proof of Corollary II.7.15

Decompose the initial operator by Theorem II.7.8 and apply the invariant packet statement to each component.

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Preservation of coherence order does not by itself give a closed classical Markov chain for the packet populations. Such scalar population equations require additional isotropy within each packet.

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II.8 Quantum error correction as a compression obstruction

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Let \(P\) be the orthogonal projector onto a code space

\[ {\mathcal C}=P{\mathcal H}, \qquad k=\dim{\mathcal C}\ge1. \]

The compression of \(A\in{\operatorname{End}}({\mathcal H})\) to the code is \(PAP\in{\operatorname{End}}({\mathcal C})\). The zero code has only the zero supported operator and no normalized code state. Part III treats that branch in the mathematical completion.

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Definition II.8.1: Code compression obstruction

Let

\[ q_P:{\operatorname{End}}({\mathcal C})\longrightarrow{\operatorname{End}}({\mathcal C})/({\mathbb C} I_{{\mathcal C}}) \]

be the quotient map. Define

\[ {\operatorname{Ob}}_P(A)=q_P(PAP). \]

Equivalently, one may use the traceless representative

\[ {\mathfrak o}_P(A) = PAP- \frac{{\operatorname{Tr}}(PAP)}{k}P. \]

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The class vanishes exactly when \(A\) acts as a scalar after compression to the code. Such an operator cannot distinguish normalized code vectors by expectation value.

Exact correction of an error span means that one completely positive trace preserving recovery \({\mathcal R}\) satisfies \({\mathcal R}(F\rho F^*)=q_F\rho\) for every \(F\) in that span and every code-supported matrix \(\rho\), 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.

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Theorem II.8.2: Knill Laflamme as obstruction vanishing

Let \(E_1,\ldots,E_r\) be an error family. The code \({\mathcal C}\) corrects their linear span exactly if and only if

\[ {\operatorname{Ob}}_P(E_a^*E_b)=0 \]

for all \(a,b\). Equivalently, there is a matrix \(c=(c_{ab})\) such that

\[ PE_a^*E_bP=c_{ab}P \]

for all \(a,b\).

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Proof of Theorem II.8.2

The equivalence between the quotient statement and the scalar compression identity is immediate from the definition. The latter identity is the finite Knill Laflamme criterion [KnillLaflamme1997], [SchumacherNielsen1996].

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The next result turns the scalar compression identities into a family of orthogonal syndrome packets. Those packets provide the coordinates used by the recovery map.

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Theorem II.8.3: Orthogonal syndrome construction

Assume the Knill Laflamme identities. Then the coefficient matrix \(c\) is positive semidefinite. Choose a unitary matrix \(W\) such that

\[ WcW^*={\operatorname{diag}}(\lambda_1,\ldots,\lambda_r), \qquad \lambda_j\geq0, \]

and define

\[ F_j=\sum_a\overline{W_{ja}}E_a. \]

For every \(j\) with \(\lambda_j>0\), the operator

\[ V_j=\lambda_j^{-1/2}F_jP \]

is an isometry from the code into \({\mathcal H}\). Its range projector

\[ Q_j=V_jV_j^* \]

is a syndrome packet, and the nonzero \(Q_j\) are pairwise orthogonal.

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Proof of Theorem II.8.3

For \(z\in{\mathbb C}^r\), let \(F_z=\sum_az_aE_a\). Then

\[ PF_z^*F_zP=(z^*cz)P\geq0, \]

so \(c\geq0\). The unitary change of error coordinates gives

\[ PF_j^*F_\ell P = \lambda_j\delta_{j\ell}P. \]

Thus

\[ V_j^*V_\ell=\delta_{j\ell}P. \]

Each \(V_j\) is an isometry on \(P{\mathcal H}\), and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.

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Theorem II.8.4: Exact recovery on the error span

Under the hypotheses of Theorem II.8.3, define on the syndrome support

\[ Q=\sum_{\lambda_j>0}Q_j \]

the completely positive map

\[ {\mathcal R}_0(X) = \sum_{\lambda_j>0} V_j^*Q_jXQ_jV_j. \]

It extends to a quantum channel \({\mathcal R}\) on \({\operatorname{End}}({\mathcal H})\). If

\[ \Phi(X)=\sum_aE_aXE_a^* \]

is trace preserving on code states, then

\[ {\mathcal R}\Phi(\rho)=\rho \]

for every density matrix \(\rho=P\rho P\).

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Proof of Theorem II.8.4

The map \({\mathcal R}_0\) is completely positive. It is trace preserving on operators supported in \(Q{\mathcal H}\) because

\[ {\operatorname{Tr}}\bigl(V_j^*Q_jXQ_jV_j\bigr) = {\operatorname{Tr}}(Q_jXQ_j). \]

Fix a code state \(\rho_0\) and write \(Q^\perp=I-Q\). Then

\[ {\mathcal R}(X) = {\mathcal R}_0(QXQ) + {\operatorname{Tr}}(Q^\perp X)\rho_0 \]

is completely positive and trace preserving. It extends \({\mathcal R}_0\) on operators supported in \(Q{\mathcal H}\).

Unitary mixing of a Kraus family does not change the channel, so

\[ \Phi(\rho) = \sum_jF_j\rho F_j^* = \sum_{\lambda_j>0} \lambda_jV_j\rho V_j^*. \]

Recovery gives

\[ {\mathcal R}\Phi(\rho) = \left(\sum_j\lambda_j\right)\rho. \]

Trace preservation on a code state implies \(\sum_j\lambda_j=1\).

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Corollary II.8.5: Syndrome measurement preserves logical information

For a channel trace preserving on code states as in the preceding theorem, the completed projective measurement consisting of \((Q_j)_{\lambda_j>0}\) and \(I-\sum_{\lambda_j>0}Q_j\) identifies the orthogonal error syndrome without resolving the encoded state inside the code. The complement outcome has probability zero on the corrupted code state.

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Proof of Corollary II.8.5

On syndrome \(j\), the encoded state is carried by the isometry \(V_j\). The outcome probability is \(\lambda_j\), independent of the normalized code state, while the postmeasurement state is \(V_j\rho V_j^*\).

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Definition II.8.6: Approximate Knill Laflamme defect

For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero.

\[ \varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in{\mathbb C}} {\left\lVert PE_a^*E_bP-cP\right\rVert}. \]

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Proposition II.8.7: Meaning of the approximate defect

For each pair \((a,b)\), choose a minimizing scalar \(c_{ab}\). For unit code vectors \(\psi,\phi\),

\[ {\left\lvert {\left\langle E_a\psi,E_b\phi\right\rangle} -c_{ab}{\left\langle\psi,\phi\right\rangle} \right\rvert} \leq \varepsilon_{\rm KL}. \]

Thus \(\varepsilon_{\rm KL}\) controls the deviation of error overlaps from their scalar code form.

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Proof of Proposition II.8.7

The left side is

\[ {\left\lvert {\left\langle\psi,(PE_a^*E_bP-c_{ab}P)\phi\right\rangle} \right\rvert}, \]

which is bounded by the operator norm.

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No recovery fidelity estimate is inferred from this norm alone. Approximate error correction requires a separate continuity theorem and a declared channel metric.

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Example II.8.8: Three qubit repetition code

Let

\[ {\left|0_L\right\rangle}={\left|000\right\rangle}, \qquad {\left|1_L\right\rangle}={\left|111\right\rangle}, \]

and take the errors

\[ I, \quad X\otimes I\otimes I, \quad I\otimes X\otimes I, \quad I\otimes I\otimes X. \]

Their images of the code are four pairwise orthogonal two dimensional syndrome packets whose sum is the full eight dimensional Hilbert space.

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Four two-dimensional syndrome packets of the repetition code. — The syndrome packets of the three qubit repetition code. Each packet has dimension two and carries the full logical state.

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Standard fact II.8.9: Correctable observable algebras

Let \(\Phi\) be a finite completely positive trace preserving channel with Kraus operators \((E_a)\). A finite dimensional unital star subalgebra \({\mathcal M}\subseteq P{\operatorname{End}}({\mathcal H})P\) is correctable when a completely positive trace preserving recovery \({\mathcal R}\) satisfies \(P(\Phi^*\circ{\mathcal R}^*)(X)P=X\) for every \(X\in{\mathcal M}\), where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product \(PE_a^*E_bP\) belongs to the commutant \({\mathcal M}'\) on the code. The scalar code criterion is the special case \({\mathcal M}=P{\operatorname{End}}({\mathcal H})P\) [BenyKempfKribs2007].

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II.9 Feshbach effective dynamics and hidden packet memory

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Let \(P\) be an orthogonal projector, put \(Q=I-P\), and write a self adjoint Hamiltonian as

\[ H= \begin{pmatrix} A&B\\ B^*&D \end{pmatrix} \]

relative to \(P{\mathcal H}\oplus Q{\mathcal H}\).

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Definition II.9.1: Feshbach map and self energy

For \(z\notin{\operatorname{Spec}} D\), define

\[ \Sigma(z)=B(z-D)^{-1}B^* \]

and

\[ F_P(z)=z-A-\Sigma(z). \]

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Theorem II.9.2: Exact projected resolvent

If \(z\notin{\operatorname{Spec}} D\) and \(F_P(z)\) is invertible, then

\[ P(z-H)^{-1}P=F_P(z)^{-1} \]

as an operator on \(P{\mathcal H}\).

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Proof of Theorem II.9.2

The lower right block of \(z-H\) is \(z-D\). Block Gaussian elimination gives

\[ \begin{pmatrix} I&B(z-D)^{-1}\\ 0&I \end{pmatrix} (z-H) \begin{pmatrix} I&0\\ (z-D)^{-1}B^*&I \end{pmatrix} = \begin{pmatrix} F_P(z)&0\\ 0&z-D \end{pmatrix}. \]

Invert the factors. The retained resolvent is the block in the upper left corner of the resulting inverse.

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Theorem II.9.3: Spectral equivalence

Let \(z\notin{\operatorname{Spec}} D\). Then

\[ z\in{\operatorname{Spec}} H \quad\Longleftrightarrow\quad 0\in{\operatorname{Spec}} F_P(z). \]

The map

\[ x\longmapsto \begin{pmatrix} x\\ (z-D)^{-1}B^*x \end{pmatrix} \]

is an isomorphism from \({\operatorname{Ker}} F_P(z)\) onto \({\operatorname{Ker}}(z-H)\).

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Proof of Theorem II.9.3

The equation \((z-H)(x,y)^{\mathsf T}=0\) is

\[ (z-A)x-By=0, \qquad -B^*x+(z-D)y=0. \]

The second equation gives \(y=(z-D)^{-1}B^*x\). Substitution into the first gives \(F_P(z)x=0\). The construction is reversible.

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Zeros of an effective Feshbach determinant align with finite eigenvalues. — The zeros of the Feshbach determinant recover the eigenvalues of the full finite Hamiltonian away from the hidden spectrum.

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Theorem II.9.4: Upper half plane sign

If \(\operatorname{Im}z>0\), then

\[ \operatorname{Im}\Sigma(z) = -(\operatorname{Im}z) B(z-D)^{-1}(\overline z-D)^{-1}B^* \leq0. \]

Thus \(-\Sigma\) is an operator valued Herglotz function.

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Proof of Theorem II.9.4

For \(R(z)=(z-D)^{-1}\), the resolvent identity gives

\[ R(z)-R(z)^* =(\overline z-z)R(z)R(z)^*. \]

Divide by \(2{\mathrm i}\), then multiply the result on the left by \(B\) and on the right by \(B^*\).

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Proposition II.9.5: Uniform self energy bounds

Let

\[ \delta={\operatorname{dist}}(z,{\operatorname{Spec}} D)>0, \qquad g={\left\lVert B\right\rVert}. \]

Then

\[ {\left\lVert\Sigma(z)\right\rVert}\leq\frac{g^2}{\delta} \]

and

\[ {\left\lVert\Sigma'(z)\right\rVert}\leq\frac{g^2}{\delta^2}. \]

For real \(E,E'\) in an interval at distance at least \(\delta\) from \({\operatorname{Spec}} D\),

\[ {\left\lVert\Sigma(E)-\Sigma(E')\right\rVert} \leq \frac{g^2}{\delta^2}{\left\lvert E-E'\right\rvert}. \]

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Proof of Proposition II.9.5

The self adjoint resolvent satisfies

\[ {\left\lVert(z-D)^{-1}\right\rVert}=\frac1{{\operatorname{dist}}(z,{\operatorname{Spec}} D)}. \]

Differentiate the resolvent to obtain

\[ \Sigma'(z)=-B(z-D)^{-2}B^*. \]

Taking operator norms gives the first two estimates. Applying the derivative bound along the real interval gives the final mean value estimate.

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Proposition II.9.6: Low energy expansion

Assume \(D\) is invertible and \({\left\lvert z\right\rvert}{\left\lVert D^{-1}\right\rVert}<1\). Then

\[ \Sigma(z) =- \sum_{n=0}^{\infty} z^nBD^{-n-1}B^* \]

with norm convergence. In particular,

\[ F_P(z) = z-A+BD^{-1}B^*+O\bigl({\left\lvert z\right\rvert}{\left\lVert B\right\rVert}^2{\left\lVert D^{-1}\right\rVert}^2\bigr). \]

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Proof of Proposition II.9.6

Write

\[ (z-D)^{-1} =-D^{-1}(I-zD^{-1})^{-1} =-\sum_{n=0}^{\infty}z^nD^{-n-1} \]

and multiply on the left by \(B\) and on the right by \(B^*\).

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Proposition II.9.7: Bare truncation error identity

If both \(F_P(z)\) and \(z-A\) are invertible, then

\[ P(z-H)^{-1}P-(z-A)^{-1} = F_P(z)^{-1}\Sigma(z)(z-A)^{-1}. \]

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Proof of Proposition II.9.7

Use the inverse difference identity

\[ X^{-1}-Y^{-1}=X^{-1}(Y-X)Y^{-1} \]

with \(X=F_P(z)\) and \(Y=z-A\).

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II.9.1 Time memory

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Write a Schrödinger state as \(\psi=x+y\), where \(x=P\psi\) and \(y=Q\psi\).

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Theorem II.9.8: Exact retained memory equation

The retained coordinate satisfies

\[ \begin{aligned} {\mathrm i}\dot x(t) ={}&Ax(t)+B{\mathrm e}^{-{\mathrm i} tD}y(0)\\ &-{\mathrm i}\int_0^tK(t-s)x(s){\,\mathrm d} s, \end{aligned} \]

where

\[ K(t)=B{\mathrm e}^{-{\mathrm i} tD}B^*. \]

For \(\operatorname{Re}s>0\),

\[ \int_0^\infty{\mathrm e}^{-st}K(t){\,\mathrm d} t = B(s+{\mathrm i} D)^{-1}B^*. \]

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Proof of Theorem II.9.8

The hidden equation is

\[ {\mathrm i}\dot y=B^*x+Dy. \]

Variation of constants gives

\[ y(t)={\mathrm e}^{-{\mathrm i} tD}y(0) -{\mathrm i}\int_0^t{\mathrm e}^{-{\mathrm i}(t-s)D}B^*x(s){\,\mathrm d} s. \]

Substitute into \({\mathrm i}\dot x=Ax+By\). The Laplace formula follows by diagonalizing \(D\) or integrating the norm convergent finite dimensional exponential.

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Theorem II.9.9: Positive definite quantum memory

The kernel \(K(t-s)\) is positive definite. For times \(t_1,\ldots,t_m\) and retained vectors \(v_1,\ldots,v_m\),

\[ \sum_{j,k=1}^{m} {\left\langle v_j,K(t_j-t_k)v_k\right\rangle} \geq0. \]

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Proof of Theorem II.9.9

Set

\[ w=\sum_{k=1}^{m}{\mathrm e}^{{\mathrm i} t_kD}B^*v_k. \]

Then

\[ {\left\lVert w\right\rVert}^2 = \sum_{j,k} {\left\langle B^*v_j,{\mathrm e}^{-{\mathrm i}(t_j-t_k)D}B^*v_k\right\rangle} = \sum_{j,k} {\left\langle v_j,K(t_j-t_k)v_k\right\rangle}. \]

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Corollary II.9.10: Finite oscillatory memory expansion

\[ D=\sum_{r}E_rQ_r \]

is the spectral decomposition of the hidden Hamiltonian, then

\[ K(t)=\sum_r{\mathrm e}^{-{\mathrm i} tE_r}BQ_rB^*. \]

Every coefficient \(BQ_rB^*\) is positive semidefinite.

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Proof of Corollary II.9.10

Inserting the spectral decomposition into the exponential gives the stated sum. The spectral projectors are positive semidefinite, so multiplication by the coupling and its adjoint preserves positivity of each coefficient.

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Real and imaginary parts of a finite oscillatory memory kernel. — A finite hidden spectrum produces an oscillatory memory kernel. The retained dynamics therefore depend on the history of the retained state.

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Theorem II.9.11: Exact local closure criterion

Assume \(y(0)=0\). The retained equation is local and equal to

\[ {\mathrm i}\dot x=Ax \]

for every retained initial state exactly when \(B=0\). Equivalently, exact local closure holds exactly when the retained and hidden packets reduce \(H\).

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Proof of Theorem II.9.11

If \(B=0\), the retained and hidden equations decouple. Conversely, suppose that every solution with \(y(0)=0\) satisfies \(x(t)={\mathrm e}^{-{\mathrm i} tA}x(0)\). The full block equation gives

\[ \ddot x(0) = -\bigl(A^2+BB^*\bigr)x(0), \]

while the declared local equation gives

\[ \ddot x(0)=-A^2x(0). \]

Since this holds for every retained initial vector, \(BB^*=0\). Hence \(B=0\).

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II.9.2 Stability across models

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Theorem II.9.12: 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

\[ \Sigma(z)=B(z-D)^{-1}B^*, \qquad \widetilde\Sigma(z)=\widetilde B(z-\widetilde D)^{-1}\widetilde B^*. \]

Assume

\[ {\left\lVert B\right\rVert},{\left\lVert\widetilde B\right\rVert}\leq g \]

and

\[ {\operatorname{dist}}(z,{\operatorname{Spec}} D), {\operatorname{dist}}(z,{\operatorname{Spec}}\widetilde D) \geq\delta>0. \]

Then

\[ {\left\lVert\Sigma(z)-\widetilde\Sigma(z)\right\rVert} \leq \frac{2g}{\delta}{\left\lVert B-\widetilde B\right\rVert} + \frac{g^2}{\delta^2}{\left\lVert D-\widetilde D\right\rVert}. \]

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Proof of Theorem II.9.12

Add and subtract \(\widetilde B(z-D)^{-1}B^*\) and \(\widetilde B(z-D)^{-1}\widetilde B^*\). The first two differences give the coupling term. For the last, use

\[ (z-D)^{-1}-(z-\widetilde D)^{-1} =(z-D)^{-1}(D-\widetilde D)(z-\widetilde D)^{-1}. \]

Apply the resolvent bounds.

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Corollary II.9.13: Uniform Feshbach control

For a family represented on fixed retained and hidden Hilbert spaces, with self adjoint hidden Hamiltonians, a common hidden spectral separation \(\delta\), and a common coupling bound \(g\), the self energies are uniformly bounded and uniformly Lipschitz on the declared spectral region. If the couplings and hidden Hamiltonians converge in norm, then the self energies converge uniformly there.

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Proof of Corollary II.9.13

The common gap and coupling bounds allow Proposition II.9.5 and Theorem II.9.12 to be applied uniformly across the family.

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This result identifies hypotheses that make Feshbach control uniform in the cutoff. Whether a chosen geometric refinement tower satisfies those hypotheses remains a separate question.

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II.10 Euclidean transfer and finite reflection positivity

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Let \(H=H^*\) on a nonzero finite dimensional Hilbert space and let

\[ E_0=\min{\operatorname{Spec}} H. \]

For a lattice time step \(a>0\), define

\[ T={\mathrm e}^{-a(H-E_0I)}. \]

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Theorem II.10.1: Finite Euclidean transfer

The operator \(T\) is positive definite, self adjoint, and contractive. Its spectrum lies in \((0,1]\), and

\[ T^n={\mathrm e}^{-na(H-E_0I)} \]

for every \(n\in{\mathbb N}\).

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Proof of Theorem II.10.1

Diagonalize \(H\). Each eigenvalue of \(T\) is

\[ {\mathrm e}^{-a(E_j-E_0)} \]

and therefore lies in \((0,1]\). The power law follows from the functional calculus.

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Theorem II.10.2: Hamiltonian reconstruction from transfer

Let \(T\) be positive definite, self adjoint, and contractive. Then

\[ H_T=-a^{-1}\log T \]

is a nonnegative self adjoint operator and

\[ T^n={\mathrm e}^{-naH_T}. \]

It is the unique nonnegative self adjoint operator with \(T={\mathrm e}^{-aH_T}\).

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Proof of Theorem II.10.2

Apply the real logarithm to the positive spectrum of \(T\). The spectral calculus gives existence and the power identity. Injectivity of the scalar exponential on \([0,\infty)\) gives uniqueness.

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Theorem II.10.3: Finite reflection positivity

For vectors \(v_1,\ldots,v_m\in{\mathcal H}\) and nonnegative integers \(n_1,\ldots,n_m\), the matrix

\[ G_{jk}={\left\langle v_j,T^{n_j+n_k}v_k\right\rangle} \]

is positive semidefinite. Equivalently,

\[ \sum_{j,k}\overline{c_j}c_k {\left\langle v_j,T^{n_j+n_k}v_k\right\rangle} \geq0 \]

for every \(c\in{\mathbb C}^m\).

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Proof of Theorem II.10.3

Since \(T\) is positive and self adjoint,

\[ {\left\langle v_j,T^{n_j+n_k}v_k\right\rangle} = {\left\langle T^{n_j}v_j,T^{n_k}v_k\right\rangle}. \]

The quadratic form is the squared norm of

\[ \sum_kc_kT^{n_k}v_k. \]

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Corollary II.10.4: Reflection positive correlators

Fix a vector \(\Omega\) and operators \(A_1,\ldots,A_m\). The matrix

\[ C_{jk} = {\left\langle A_j\Omega,T^{n_j+n_k}A_k\Omega\right\rangle} \]

is positive semidefinite.

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Proof of Corollary II.10.4

Apply Theorem II.10.3 with \(v_j=A_j\Omega\).

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Eigenvalues of a finite reflection-positive Gram matrix. — Eigenvalues of a finite reflection positive Gram matrix. Small numerical negative values would signal either roundoff or failure of the transfer assumptions.

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Proposition II.10.5: Ground packet convergence

Let \(P_0\) be the spectral projector of \(H\) at \(E_0\), and let

\[ \Delta=\min\{E_j-E_0:E_j>E_0\}. \]

If an excited eigenvalue exists, then

\[ {\left\lVert T^n-P_0\right\rVert}={\mathrm e}^{-na\Delta}. \]

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Proof of Proposition II.10.5

In the spectral decomposition, \(T^n-P_0\) has eigenvalues zero on the ground space and \({\mathrm e}^{-na(E_j-E_0)}\) on the excited eigenspaces. The largest is \({\mathrm e}^{-na\Delta}\).

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Proposition II.10.6: Symmetry resolved transfer

If \([H,P_a]=0\) for a complete packet family, then

\[ T=\sum_aP_aTP_a \]

and each reflection positive Gram matrix decomposes as the sum of its packet Gram matrices.

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Proof of Proposition II.10.6

The functional calculus preserves commutation. Decompose every vector into packet components and use orthogonality.

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Finite reflection positivity is a necessary structural property of transfer data. It is not by itself the continuum Osterwalder Schrader theorem, which also requires a suitable algebra of Euclidean fields, covariance, regularity, clustering, and a controlled continuum limit [OsterwalderSchrader1973], [OsterwalderSchrader1975], [GlimmJaffe1987].

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II.11 Finite spectral geometry and diffusion dimension

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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 [Connes1994], [Vassilevich2003].

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Definition II.11.1: Heat trace and spectral dimension

Let the eigenvalues of \(L\) be \(\lambda_1,\ldots,\lambda_N\), with at least one zero eigenvalue. Define

\[ Z_L(t)={\operatorname{Tr}}({\mathrm e}^{-tL}) \]

and

\[ d_L(t) =-2\frac{{\,\mathrm d}\log Z_L(t)}{{\,\mathrm d}\log t} = 2t \frac{{\operatorname{Tr}}(L{\mathrm e}^{-tL})}{{\operatorname{Tr}}({\mathrm e}^{-tL})}. \]

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Proposition II.11.2: Finite scale limits

For every finite nonnegative \(L\) with nontrivial kernel,

\[ \lim_{t\downarrow0}d_L(t)=0 \]

and

\[ \lim_{t\to\infty}d_L(t)=0. \]

For every \(t>0\),

\[ 0\leq d_L(t)\leq2t{\left\lVert L\right\rVert}. \]

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Proof of Proposition II.11.2

The formula in Definition II.11.1 is twice \(t\) times a weighted average of the eigenvalues. This gives the bound and the limit at zero. At large time, the denominator converges to \(\dim{\operatorname{Ker}} L\), while every positive eigenvalue contribution in the numerator decays exponentially.

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Thus a finite system cannot have a nonzero spectral dimension at the extreme ultraviolet or infrared ends. A geometric dimension can appear only in a refinement window.

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Proposition II.11.3: Heat packet decomposition

If \([L,P_a]=0\) for every packet, then

\[ Z_L(t)=\sum_aZ_{L,a}(t), \qquad Z_{L,a}(t)={\operatorname{Tr}}(P_a{\mathrm e}^{-tL}). \]

The global spectral dimension is the average of the packet dimensions weighted by their heat traces:

\[ d_L(t) = \sum_{a:Z_{L,a}(t)>0} \frac{Z_{L,a}(t)}{Z_L(t)} d_{L,a}(t), \]

where

\[ d_{L,a}(t) =2t \frac{{\operatorname{Tr}}(P_aL{\mathrm e}^{-tL})}{Z_{L,a}(t)} \]

for nonempty packet heat traces.

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Proof of Proposition II.11.3

Commutation gives a block diagonal heat operator. Add the traces and the corresponding first spectral moments, then divide by the total heat trace.

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II.11.1 Periodic lattices

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Let \(N\geq2\) and let \(\Lambda_N=({\mathbb Z}/N{\mathbb Z})^d\) be the unit flat torus grid with spacing \(1/N\). The standard periodic difference Laplacian has eigenvalues

\[ \lambda_N(k) = 4N^2 \sum_{j=1}^{d} \sin^2\left(\frac{\pi k_j}{N}\right), \]

where \(k\) runs through centered residue representatives. This is the standard periodic finite difference Laplacian [Strikwerda2004].

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Theorem II.11.4: Low mode dispersion in dimension \(d\)

For every centered mode \(k\),

\[ 0 \leq 4\pi^2{\left\lvert k\right\rvert}^2-\lambda_N(k) \leq \frac{4\pi^4}{3N^2} \sum_{j=1}^{d}k_j^4. \]

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Proof of Theorem II.11.4

For \(x\in[0,\pi/2]\),

\[ \sin x\leq x \]

and

\[ \sin x\geq x-\frac{x^3}{6}. \]

Thus

\[ 0\leq x^2-\sin^2x\leq\frac{x^4}{3}. \]

Apply this with \(x=\pi{\left\lvert k_j\right\rvert}/N\), multiply by \(4N^2\), and sum over \(j\).

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Theorem II.11.5: Fixed time heat trace convergence

For every \(t>0\),

\[ \sum_{k\in\Lambda_N^*} {\mathrm e}^{-t\lambda_N(k)} \longrightarrow \sum_{k\in{\mathbb Z}^d} {\mathrm e}^{-4\pi^2t{\left\lvert k\right\rvert}^2} \]

as \(N\to\infty\). The first spectral moments also converge:

\[ \sum_{k\in\Lambda_N^*} \lambda_N(k){\mathrm e}^{-t\lambda_N(k)} \longrightarrow \sum_{k\in{\mathbb Z}^d} 4\pi^2{\left\lvert k\right\rvert}^2{\mathrm e}^{-4\pi^2t{\left\lvert k\right\rvert}^2}. \]

Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.

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Proof of Theorem II.11.5

For a fixed integer mode, Theorem II.11.4 gives

\[ \lambda_N(k)\longrightarrow4\pi^2{\left\lvert k\right\rvert}^2. \]

For centered modes, the inequality

\[ \sin x\geq\frac{2x}{\pi}, \qquad 0\leq x\leq\frac\pi2, \]

gives

\[ \lambda_N(k)\geq16{\left\lvert k\right\rvert}^2. \]

Also \(\lambda_N(k)\leq4\pi^2{\left\lvert k\right\rvert}^2\). Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by

\[ {\mathrm e}^{-16t{\left\lvert k\right\rvert}^2}, \]

and the moment terms by

\[ 4\pi^2{\left\lvert k\right\rvert}^2{\mathrm e}^{-16t{\left\lvert k\right\rvert}^2}. \]

Both dominating sequences are summable on \({\mathbb Z}^d\). Dominated convergence proves the two limits. Their quotient gives spectral dimension convergence.

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Standard fact II.11.6: Theta transformation

The continuum flat torus heat trace satisfies

\[ Z_d(t) = \sum_{k\in{\mathbb Z}^d}{\mathrm e}^{-4\pi^2t{\left\lvert k\right\rvert}^2} = (4\pi t)^{-d/2} \sum_{n\in{\mathbb Z}^d}{\mathrm e}^{-{\left\lvert n\right\rvert}^2/(4t)}. \]

This is the Poisson summation formula for a Gaussian.

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Theorem II.11.7: Continuum torus small time dimension

Let

\[ d_{\rm torus}(t) =-2\frac{{\,\mathrm d}\log Z_d(t)}{{\,\mathrm d}\log t}. \]

Then

\[ d_{\rm torus}(t) = d+O\left(t^{-1}{\mathrm e}^{-1/(4t)}\right) \]

as \(t\downarrow0\).

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Proof of Theorem II.11.7

By Standard fact II.11.6,

\[ Z_d(t) =(4\pi t)^{-d/2}(1+R(t)), \]

where

\[ R(t)=\sum_{n\neq0}{\mathrm e}^{-{\left\lvert n\right\rvert}^2/(4t)} =O\bigl({\mathrm e}^{-1/(4t)}\bigr). \]

Differentiation gives

\[ tR'(t)=O\left(t^{-1}{\mathrm e}^{-1/(4t)}\right). \]

Substituting this estimate into the logarithmic derivative gives the stated small time expansion.

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Corollary II.11.8: Rigorous intermediate dimension window

For every \(\varepsilon>0\), there is a diffusion time \(t_\varepsilon>0\) and an integer \(N_\varepsilon\) such that

\[ {\left\lvert d_{L_N}(t_\varepsilon)-d\right\rvert}<\varepsilon \]

for every \(N\geq N_\varepsilon\).

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Proof of Corollary II.11.8

Choose \(t_\varepsilon\) so that Theorem II.11.7 is within \(\varepsilon/2\) of \(d\). Then apply Theorem II.11.5 to choose \(N_\varepsilon\) so that the discrete spectral dimension is within \(\varepsilon/2\) of the continuum value.

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Intermediate spectral-dimension windows for finite periodic lattices. — Finite periodic lattices show an intermediate dimension window. The ultraviolet and infrared limits return to zero because the state space is finite.

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The order of limits matters. Taking \(t\downarrow0\) at fixed \(N\) gives zero. Taking \(N\to\infty\) at fixed positive \(t\), then taking \(t\downarrow0\), gives the continuum dimension.

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II.12 Constraint closure and anomaly packets

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Let \(C_1,\ldots,C_m\) 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 [Dirac1964], [HenneauxTeitelboim1992]. Write

\[ \mathfrak c=\operatorname{span}_{{\mathbb C}}\{C_1,\ldots,C_m\} \subseteq{\operatorname{End}}({\mathcal H}). \]

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Definition II.12.1: Universal closure obstruction

Let

\[ q_{\mathfrak c}:{\operatorname{End}}({\mathcal H})\longrightarrow{\operatorname{End}}({\mathcal H})/\mathfrak c \]

be the quotient map. Define

\[ {\operatorname{Ob}}_{ab}^{\rm cl} = q_{\mathfrak c}([C_a,C_b]). \]

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Theorem II.12.2: Finite constraint closure criterion

The constraint span \(\mathfrak c\) is closed under commutators exactly when

\[ {\operatorname{Ob}}_{ab}^{\rm cl}=0 \]

for all \(a,b\).

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Proof of Theorem II.12.2

The quotient class vanishes exactly when \([C_a,C_b]\in\mathfrak c\). This holds for every pair exactly when \(\mathfrak c\) is a Lie algebra under the commutator.

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If structure coefficients \(f_{ab}^{\phantom{ab}c}\) are prescribed, define the fixed structure anomaly

\[ \mathfrak A_{ab} = [C_a,C_b] -{\mathrm i}\sum_cf_{ab}^{\phantom{ab}c}C_c. \]

Exact realization of the prescribed algebra is the stronger condition \(\mathfrak A_{ab}=0\).

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Proposition II.12.3: Physical compression is not a closure test

Let

\[ {\mathcal H}_{{\mathrm{phys}}}=\bigcap_a{\operatorname{Ker}} C_a \]

and let \(P_{{\mathrm{phys}}}\) be its orthogonal projector. Then

\[ P_{{\mathrm{phys}}}[C_a,C_b]P_{{\mathrm{phys}}}=0 \]

for every pair \(a,b\), whether or not the constraint span closes.

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Proof of Proposition II.12.3

Self adjointness and \(C_aP_{{\mathrm{phys}}}=0\) give \(P_{{\mathrm{phys}}}C_a=0\). Both terms in the compressed commutator therefore vanish.

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The quotient in Definition II.12.1 is needed because closure concerns the operator algebra. Compression to physical states tests only the action on the common kernel, where the constraints already vanish.

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Constraint commutators are tested in a quotient by the constraint span. — The closure obstruction lives in a quotient by the constraint span. Physical compression is automatic for self adjoint constraints and cannot replace this quotient.

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Theorem II.12.4: Invariance of the physical space

Let \(H=H^*\). Suppose

\[ [H,C_a] = \sum_cb_a^{\phantom a c}C_c \]

for every \(a\). Then \(H\) preserves \({\mathcal H}_{{\mathrm{phys}}}\), and so does \({\mathrm e}^{-{\mathrm i} tH}\) for every real \(t\).

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Proof of Theorem II.12.4

For \(\psi\in{\mathcal H}_{{\mathrm{phys}}}\),

\[ C_aH\psi =HC_a\psi-[H,C_a]\psi =0. \]

Thus \(H\psi\in{\mathcal H}_{{\mathrm{phys}}}\). The exponential preserves every invariant finite dimensional subspace.

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II.12.1 Constraint refinement

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Let \(({\mathcal H}_\ell,C_{\ell,a})\) and \(({\mathcal H}_{\ell+1},C_{\ell+1,a})\) be two levels, and let

\[ R_\ell:{\mathcal H}_\ell\longrightarrow{\mathcal H}_{\ell+1} \]

be an isometry.

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Theorem II.12.5: Exact constraint transport

Assume

\[ C_{\ell+1,a}R_\ell=R_\ell C_{\ell,a} \]

for every \(a\). Then

\[ [C_{\ell+1,a},C_{\ell+1,b}]R_\ell = R_\ell[C_{\ell,a},C_{\ell,b}]. \]

If the same structure coefficients are used at both levels, then

\[ \mathfrak A_{\ell+1,ab}R_\ell = R_\ell\mathfrak A_{\ell,ab}. \]

The isometry maps coarse physical states into fine physical states.

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Proof of Theorem II.12.5

Apply the intertwining identity twice to each product in the commutator. The anomaly identity follows by subtracting the same linear combination of constraints. If \(C_{\ell,a}\psi=0\), then

\[ C_{\ell+1,a}R_\ell\psi=R_\ell C_{\ell,a}\psi=0. \]

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Proposition II.12.6: Approximate constraint transport

Assume

\[ {\left\lVert C_{\ell+1,a}R_\ell-R_\ell C_{\ell,a}\right\rVert} \leq\varepsilon_a \]

and

\[ {\left\lVert C_{\ell,a}\right\rVert},{\left\lVert C_{\ell+1,a}\right\rVert}\leq M. \]

Then a coarse physical unit vector \(\psi\) satisfies

\[ {\left\lVert C_{\ell+1,a}R_\ell\psi\right\rVert}\leq\varepsilon_a. \]

Also,

\[ \begin{aligned} &{\left\lVert [C_{\ell+1,a},C_{\ell+1,b}]R_\ell -R_\ell[C_{\ell,a},C_{\ell,b}] \right\rVert}\\ &\hspace{4em}\leq 2M(\varepsilon_a+\varepsilon_b). \end{aligned} \]

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Proof of Proposition II.12.6

The first claim follows by applying the intertwining defect to \(\psi\). Write

\[ E_a=C_{\ell+1,a}R_\ell-R_\ell C_{\ell,a}. \]

Expanding the two commutators gives four error terms:

\[ E_aC_{\ell,b}, \quad C_{\ell+1,a}E_b, \quad -E_bC_{\ell,a}, \quad -C_{\ell+1,b}E_a. \]

The triangle inequality and the stated bounds on the constraints and defects give the claimed estimate.

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Approximate transport controls finite violations. It does not turn a nonzero anomaly into an exact continuum algebra without a limit theorem and a common operator domain.

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II.13 Quantum refinement towers

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Definition II.13.1: Finite quantum refinement tower

A finite quantum refinement tower consists of Hilbert spaces \({\mathcal H}_\ell\), isometries

\[ R_\ell:{\mathcal H}_\ell\longrightarrow{\mathcal H}_{\ell+1}, \]

self adjoint Hamiltonians \(H_\ell\), packet projectors \(P_{\ell,a}\), constraints \(C_{\ell,r}\), and positive transfer operators \(T_\ell\). Exact compatibility means that every declared object intertwines with \(R_\ell\). Approximate compatibility is recorded by the corresponding operator norm defects.

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Four refinement levels carry Hamiltonians, packet projectors, and constraints. — A refinement tower carries Hamiltonians, packets, constraints, and transfer data. Each family has its own compatibility defect.

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II.13.1 Exact bounded direct limits

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Let \({\mathcal H}_\infty\) be the Hilbert direct limit of the isometric system, and let

\[ \iota_\ell:{\mathcal H}_\ell\longrightarrow{\mathcal H}_\infty \]

be the canonical isometries.

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Theorem II.13.2: Bounded Hamiltonian direct limit

Assume

\[ H_{\ell+1}R_\ell=R_\ell H_\ell \]

and

\[ \sup_\ell{\left\lVert H_\ell\right\rVert}<\infty. \]

Then there is a unique bounded self adjoint operator \(H_\infty\) on \({\mathcal H}_\infty\) such that

\[ H_\infty\iota_\ell=\iota_\ell H_\ell \]

for every \(\ell\). It satisfies

\[ {\mathrm e}^{-{\mathrm i} tH_\infty}\iota_\ell = \iota_\ell{\mathrm e}^{-{\mathrm i} tH_\ell} \]

and

\[ {\mathrm e}^{-tH_\infty}\iota_\ell = \iota_\ell{\mathrm e}^{-tH_\ell} \]

when the Hamiltonians are nonnegative.

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Proof of Theorem II.13.2

Define \(H_\infty\) on the algebraic union of the embedded levels by

\[ H_\infty\iota_\ell x=\iota_\ell H_\ell x. \]

Compatibility makes the definition independent of the chosen level. The uniform norm bound makes the operator bounded on the dense union, so it extends uniquely to the completion. It is symmetric because every \(H_\ell\) is self adjoint. A bounded symmetric operator defined on the whole Hilbert space is self adjoint. The functional calculus intertwining follows first for polynomials and then for exponentials by uniform convergence.

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Theorem II.13.3: 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

\[ P_{\ell+1,a}R_\ell=R_\ell P_{\ell,a}. \]

Then there is an orthogonal projector \(P_{\infty,a}\) on \({\mathcal H}_\infty\) satisfying

\[ P_{\infty,a}\iota_\ell=\iota_\ell P_{\ell,a}. \]

Finite completeness and orthogonality pass to the direct limit.

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Proof of Theorem II.13.3

Use the same construction as in Theorem II.13.2. Every projector has norm at most one, so no additional boundedness assumption is needed. The projector identities hold on the dense algebraic union and hence on the completion.

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II.13.2 Norm summable approximation

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The next theorem places all approximating Hamiltonians on a common Hilbert space. It applies to bounded effective Hamiltonians. An unbounded field Hamiltonian requires a different convergence argument.

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Theorem II.13.4: Norm summable Hamiltonian tower

Let \((H_n)\) be bounded self adjoint operators on one Hilbert space and assume

\[ \sum_{n=0}^{\infty}{\left\lVert H_{n+1}-H_n\right\rVert}<\infty. \]

Then \(H_n\) converges in norm to a bounded self adjoint operator \(H\). For every \(T>0\),

\[ \sup_{{\left\lvert t\right\rvert}\leq T} {\left\lVert{\mathrm e}^{-{\mathrm i} tH_n}-{\mathrm e}^{-{\mathrm i} tH}\right\rVert} \leq T{\left\lVert H_n-H\right\rVert}. \]

If every \(H_n\geq0\), then

\[ \sup_{0\leq t\leq T} {\left\lVert{\mathrm e}^{-tH_n}-{\mathrm e}^{-tH}\right\rVert} \leq T{\left\lVert H_n-H\right\rVert}. \]

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Proof of Theorem II.13.4

The summability condition makes \((H_n)\) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives

\[ {\mathrm e}^{-{\mathrm i} tH_n}-{\mathrm e}^{-{\mathrm i} tH} =-{\mathrm i}\int_0^t {\mathrm e}^{-{\mathrm i}(t-s)H_n}(H_n-H){\mathrm e}^{-{\mathrm i} sH}{\,\mathrm d} s. \]

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.

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Corollary II.13.5: Limit of packet leakage

Let \(P_n\) be orthogonal projectors and \(H_n\) bounded self adjoint operators on one Hilbert space, with \(P_n\to P\) and \(H_n\to H\) in norm. Then for every bounded time interval,

\[ (I-P_n){\mathrm e}^{-{\mathrm i} tH_n}P_n \longrightarrow (I-P){\mathrm e}^{-{\mathrm i} tH}P \]

uniformly in time.

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Proof of Corollary II.13.5

Add and subtract terms so that one factor changes at a time. The projector terms converge uniformly because \(P_n\to P\) in norm. Duhamel's identity gives

\[ \sup_{{\left\lvert t\right\rvert}\leq T} {\left\lVert{\mathrm e}^{-{\mathrm i} tH_n}-{\mathrm e}^{-{\mathrm i} tH}\right\rVert} \leq T{\left\lVert H_n-H\right\rVert}, \]

which proves uniform convergence of the middle factor.

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Theorem II.13.6: Reflection positivity is closed under limits

Suppose \(C_n(j,k)\) are reflection positive kernels on \({\mathbb N}\), meaning every finite matrix

\[ \bigl(C_n(j_r,j_s)\bigr)_{r,s} \]

is positive semidefinite. If

\[ C_n(j,k)\longrightarrow C(j,k) \]

for every pair, then \(C\) is reflection positive.

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Proof of Theorem II.13.6

For a fixed finite set of times and coefficients, the corresponding quadratic forms are nonnegative and converge to the quadratic form of \(C\). The limit is nonnegative.

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Theorem II.13.7: Constraint closure under norm convergence

Let \(C_{n,a}\to C_a\) in norm for every \(a\), and fix structure coefficients \(f_{ab}^{\phantom{ab}c}\). If

\[ {\left\lVert [C_{n,a},C_{n,b}] -{\mathrm i}\sum_cf_{ab}^{\phantom{ab}c}C_{n,c} \right\rVert} \longrightarrow0, \]

then

\[ [C_a,C_b] ={\mathrm i}\sum_cf_{ab}^{\phantom{ab}c}C_c. \]

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Proof of Theorem II.13.7

Operator multiplication and the commutator are continuous in operator norm. Pass to the limit in the anomaly identity.

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Theorem II.13.8: 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

\[ {\mathrm e}^{-tL_n}\longrightarrow{\mathrm e}^{-tL} \]

in trace norm. Then

\[ {\operatorname{Tr}}({\mathrm e}^{-tL_n})\longrightarrow{\operatorname{Tr}}({\mathrm e}^{-tL}). \]

If also the heat moment operators are trace class and

\[ L_n{\mathrm e}^{-tL_n}\longrightarrow L{\mathrm e}^{-tL} \]

in trace norm and the limiting heat trace is nonzero, then the spectral dimensions defined by the heat moment ratio converge at that time.

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Proof of Theorem II.13.8

The trace is continuous in trace norm. Apply this to the heat operators and to the first heat moments, then take the quotient in Definition II.11.1.

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Remark II.13.9

The theorems above produce bounded continuum Hamiltonians. A physically useful continuum Hamiltonian is often unbounded. Its construction generally requires strong resolvent convergence, convergence of closed quadratic forms, or another method that controls operator domains [Kato1995], [ReedSimon1972]. None of those conclusions follows from finite matrix convergence without additional hypotheses.

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II.14 Continuum milestones and exact boundary

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The finite theorems establish exact identities and criteria for bounded limits. Passing from a finite transfer model to a continuum theory still requires several separate arguments. This section states those remaining mathematical tasks.

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Definition II.14.1: Anomaly free continuum constraint system

A continuum constraint system consists of a Hilbert space, a common invariant dense domain \({\mathcal D}\), and operators \(C_a\) on \({\mathcal D}\). It is anomaly free for prescribed structure coefficients when

\[ [C_a,C_b]\psi = {\mathrm i}\sum_cf_{ab}^{\phantom{ab}c}C_c\psi \]

for every \(\psi\in{\mathcal D}\), and when the associated gauge evolution is defined on the physical quotient or physical subspace.

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Norm closure of bounded finite constraints, as in Theorem II.13.7, is one special case. It does not treat unbounded constraints, changing domains, structure functions, or renormalized operator products.

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Definition II.14.2: Reflection positive Euclidean limit

A reflection positive Euclidean limit requires a limiting algebra of observables, Euclidean time translations, a reflection operation, and correlation functions whose reflected quadratic forms are nonnegative. A field theoretic reconstruction also requires the regularity and covariance assumptions needed to construct a Hilbert space, a vacuum, and a self adjoint time translation generator.

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Theorem II.13.6 shows that positivity of each fixed finite Gram matrix survives pointwise convergence. It does not prove existence, locality, clustering, or the remaining Osterwalder Schrader axioms.

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Definition II.14.3: 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

\[ \Gamma_{\rm eff}[g] = \int \sqrt{{\left\lvert g\right\rvert}} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) {\,\mathrm d}^dx. \]

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 estimate for the omitted terms on the declared infrared regime.

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A spectral dimension close to four does not imply this criterion. A heat trace records eigenvalue growth. It does not determine the tensor structure, gauge identities, sign, or interaction coefficients of an effective action.

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Definition II.14.4: Massless spin two criterion

A positive massless spin two sector requires a physical two point function or resolvent with a massless pole in the transverse traceless sector, positive residue on physical states, absence of negative norm modes, and compatibility with the limiting gauge constraints. Interactions must preserve the constraint structure.

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A low energy eigenvalue, a spectral dimension plateau, or a finite rank two tensor packet does not establish a massless spin two particle.

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A staircase lists separate theorems required beyond finite packet identities. — The finite identities occupy the first step. Every later step needs a separate convergence or reconstruction theorem.

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What the finite results establish and what remains outside the article.
Milestone	Finite result available here	Missing continuum input
Symmetry and constraints	exact packets and quotient obstruction	common unbounded domain and renormalized closure
Effective dynamics	exact Feshbach map and uniform gap bounds	cutoff independent spectral separation and locality
Euclidean transfer	positive finite transfer and finite reflection positivity	convergent Euclidean fields and reconstruction axioms
Spectral dimension	lattice heat trace convergence	relation to a physical metric and interacting observables
Lorentzian theory	bounded unitary direct limits	unbounded generator, causality, locality, and field algebra
Einstein regime	a definition of the required expansion	derivation and error control for the effective action
Spin two sector	a precise spectral criterion	positive physical pole and interaction consistency

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II.15 Conclusion

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A complete packet family gives one coordinate language for finite quantum structure. Group characters produce symmetry packets. Charged operators obey exact selection rules. Circuits become coherent sums over packet paths. The canonical moving connection separates geometric transport from physical mixing. Density matrices and channels use two sided packets, while channel covariance becomes block preservation on operator space and symmetry of the Choi matrix.

Compression to a code gives a natural error correction obstruction. Its vanishing is the Knill Laflamme condition. Diagonalizing the resulting coefficient matrix produces orthogonal syndrome packets and an exact recovery on the error span. Schur Feshbach elimination gives an equally exact description of hidden Hamiltonian packets. Their frequency response is the self energy, and their time response is a positive definite memory kernel.

Positive Euclidean transfer gives finite reflection positivity. Periodic lattice heat traces converge to the continuum torus at every fixed positive diffusion time, and the continuum small time spectral dimension is recovered after the refinement limit. Constraint closure is measured by a quotient of operator space, not by compression to the physical kernel.

The theorems for bounded towers show how exact compatibility and summable differences in operator norm produce limiting quantum evolutions. Their hypotheses also mark the scope of the argument. Unbounded field Hamiltonians, continuum gauge constraints, reflection positive field measures, the Einstein infrared action, and a massless spin two sector each require additional theorems. The finite packet calculus gives a common framework in which those questions can be stated and studied, while leaving their physical resolution open.

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II.1 Introduction

II.2 Quantum seeds and observable algebras

Definition II.2.1: Finite quantum seed system

Theorem II.2.2: Orbit dimension of one seed

Proof of Theorem II.2.2

Theorem II.2.3: Quantum seed rank

Proof of Theorem II.2.3

Corollary II.2.4: Local orbit and Schmidt rank

Proof of Corollary II.2.4

Proposition II.2.5: Computational address orbit

Proof of Proposition II.2.5

Remark II.2.6

II.3 Symmetry packets and selection rules

Theorem II.3.1: Finite group isotypic projectors

Proof of Theorem II.3.1

Theorem II.3.2: Representation matrix units

Proof of Theorem II.3.2

II.3.1 Cyclic charge packets

Corollary II.3.3: Cyclic charge projectors

Proof of Corollary II.3.3

Definition II.3.4: Operator charge

Theorem II.3.5: Charge selection rule

Proof of Theorem II.3.5

Corollary II.3.6: Charge addition

Proof of Corollary II.3.6

II.3.2 Tensor refinement

Theorem II.3.7: Tensor charge refinement

Proof of Theorem II.3.7

Corollary II.3.8: Tensor selection rule

Proof of Corollary II.3.8

Remark II.3.9

II.4 Quantum circuits as packet paths

Theorem II.4.1: Exact circuit path expansion

Proof of Theorem II.4.1

Theorem II.4.2: Packet unitarity identities

Proof of Theorem II.4.2

Corollary II.4.3: Probability conservation from one packet

Proof of Corollary II.4.3

Theorem II.4.4: Path interference formula

Proof of Theorem II.4.4

Proposition II.4.5: Charge pruning of circuit paths

Proof of Proposition II.4.5

Example II.4.6: Bell circuit

II.5 Symmetry resolved dynamics and leakage

Theorem II.5.1: Exact invariant packets

Proof of Theorem II.5.1

Lemma II.5.2: Off diagonal norm

Proof of Lemma II.5.2

Theorem II.5.3: Fixed packet leakage

Proof of Theorem II.5.3

Corollary II.5.4: Leakage probability

Proof of Corollary II.5.4

II.6 Moving packets and geometric transport

Theorem II.6.1: Canonical packet connection

Proof of Theorem II.6.1

Theorem II.6.2: Exact Kato transport

Proof of Theorem II.6.2

Theorem II.6.3: Moving frame Hamiltonian

Proof of Theorem II.6.3

Theorem II.6.4: Moving packet leakage

Proof of Theorem II.6.4

II.6.1 Discrete packet transport

Theorem II.6.5: Direct packet rotation

Proof of Theorem II.6.5

Theorem II.6.6: Convergence of short packet rotations

Proof of Theorem II.6.6

II.6.2 Local frames and curvature

Theorem II.6.7: Nonabelian packet connection

Proof of Theorem II.6.7

Corollary II.6.8: Bloch sphere curvature

Proof of Corollary II.6.8

II.7 Operator packets, measurements, and quantum channels

Definition II.7.1: Two sided operator packet

Theorem II.7.2: Complete operator packet system

Proof of Theorem II.7.2

Definition II.7.3: Packet dephasing