Realization Calculus

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1 \section{Quantitative leakage from the actual exponential}\label{supp4:section:001}
2 Let $A$ be a unital complex C-star algebra, let $H=H^*$, and let $P=P^*=P^2$. Put $Q=1-P$. The checked statement is
3 \[
4 \boxed{\|Q\exp(-itH)P\|\le |t|\,\|HP-PH\|\qquad(t\in\R).}
5 \]
6 The matrix specialization uses the operator norm on the Hilbert space, implemented by \path{Matrix.Norms.L2Operator}. It is not an entrywise norm or the row-sum operator norm. The theorem includes the zero projector and empty matrix index type. The theorem also applies at time zero.
7 
8 \subsection{Deriving the estimate by differentiation}
9 For a skew-adjoint generator $G$, define the actual conjugation path
10 \[
11 T_P(t)=\exp(tG)P\exp(-tG).
12 \]
13 The formalization derives its derivative from the exponential power series and the product rule:
14 \[
15 T_P'(t)=\exp(tG)(GP-PG)\exp(-tG).
16 \]
17 Both exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore
18 \[
19 \|T_P'(t)\|=\|GP-PG\|,
20 \qquad
21 \|T_P(t)-T_P(s)\|\le |t-s|\,\|GP-PG\|.
22 \]
23 The second inequality follows from the mean value inequality in normed spaces and covers either ordering of $s$ and $t$. It is not a finite-difference estimate or a numerical time discretization.
24 
25 Multiplication by $\exp(tG)$ gives
26 \[
27 (T_P(t)-P)\exp(tG)=\exp(tG)P-P\exp(tG).
28 \]
29 For every orthogonal projector, $\|P\|\le1$ and $\|1-P\|\le1$. The exact corner identity
30 \[
31 (1-P)UP=(1-P)(UP-PU)P
32 \]
33 then bounds leakage by the exponential commutator. Taking $G=-iH$ gives the stated theorem because $\|[G,P]\|=\|[H,P]\|$.
34 
35 \subsection{Checked consequences}
36 Unitarity also gives the capped estimate
37 \[
38 \|Q\exp(-itH)P\|\le\min\{1,\ |t|\,\|[H,P]\|\}.
39 \]
40 If $H=H_0+V$, both summands are self-adjoint, and $[H_0,P]=0$, then
41 \[
42 \|Q\exp(-it(H_0+V))P\|\le2|t|\,\|V\|.
43 \]
44 The latter uses $\|[V,P]\|\le2\|V\|$. Its constant is not claimed to be optimal. Exact commutation gives zero leakage. The work does not include unbounded generators or time-dependent Hamiltonians.
45 
46 
47 
48 \section{Actual projected differentiation and curvature}\label{supp4:section:002}
49 Let $E,F$ be real normed spaces. Let $P:E\to\mathcal L(F,F)$ be differentiable with $P(x)^2=P(x)$ for every $x$. Let $s:E\to F$ be $C^2$ with $P(x)s(x)=s(x)$, and let $X,Y:E\to E$ be differentiable vector fields. The checked connection is the actual expression
50 \[
51 \nabla_Xs(x)=P(x)\,Ds(x)[X(x)].
52 \]
53 It is proved to preserve the range condition. It is linear in the vector field direction and obeys the scalar Leibniz rule
54 \[
55 \nabla_X(cs)=Dc[X]\,s+c\nabla_Xs
56 \]
57 for differentiable real scalar functions $c$. No differentiated projector identities are inserted as hypotheses in the main curvature theorem.
58 
59 \subsection{Differentiated projector and section identities}
60 Differentiating $P^2=P$ and $Ps=s$ produces
61 \begin{align*}
62 DP[v]P+PDP[v]&=DP[v],\\
63 PDP[v]P&=0,\\
64 Ds[v]&=DP[v]s+P Ds[v].
65 \end{align*}
66 The zero sandwich follows algebraically from the first identity. All products preserve their noncommutative order.
67 
68 \subsection{Retaining the bracket of vector fields}
69 Define the actual curvature action by
70 \[
71 R_P(X,Y)s=\nabla_X\nabla_Ys-\nabla_Y\nabla_Xs-\nabla_{[X,Y]}s.
72 \]
73 The source uses the Lie bracket defined through Fr\'echet derivatives. Symmetry of second derivatives is supplied by the $C^2$ hypothesis, not postulated separately. Expanding the connection gives
74 \[
75 R_P(X,Y)s=P\bigl(DP[X]Ds[Y]-DP[Y]Ds[X]\bigr).
76 \]
77 Substituting the section identity and removing the zero sandwich terms proves
78 \[
79 \boxed{R_P(X,Y)s=P\bigl(DP[X]DP[Y]-DP[Y]DP[X]\bigr)P s.}
80 \]
81 Thus the coefficient is precisely $P(\dd P\wedge\dd P)P$ with the convention
82 $(\dd P\wedge\dd P)(v,w)=DP[v]DP[w]-DP[w]DP[v]$.
83 
84 \subsection{Tensor properties and the remaining global boundary}
85 The coefficient is proved bilinear, alternating, and supported on the range on both sides. The chain rule proves the pullback identity
86 \[
87 F_{P\circ f}(x)(v,w)=F_P(f(x))(Df(x)v,Df(x)w).
88 \]
89 The main theorem has global regularity hypotheses on normed parameter spaces. The pullback identity alone does not supply localization, smooth local trivializations, or gluing. The final supplement constructs the global range bundle and proves that the local connections glue.
90 
91 
92 
93 \section{A constructive and total syndrome recovery}\label{supp4:section:003}
94 Let $P$ be an orthogonal code projector. The input certificate consists of a finite family $V_a$ satisfying
95 \[
96 V_aP=V_a,
97 \qquad
98 V_a^*V_b=\delta_{ab}P.
99 \]
100 These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.
101 
102 \subsection{Constructing the recovery on the whole ambient space}
103 Define the syndrome projectors and their complement by
104 \[
105 S_a=V_aV_a^*,\qquad S=\sum_aS_a,\qquad Q=1-S.
106 \]
107 The source proves self-adjointness, $S_aS_b=\delta_{ab}S_a$, $S^2=S$, and $Q^2=Q=Q^*$. It also proves $QV_a=0$ and $V_a^*Q=0$.
108 
109 The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are $R_a=V_a^*$ together with $R_{\mathrm{extra}}=Q$, giving
110 \[
111 \mathcal R(\rho)=\sum_aV_a^*\rho V_a+Q\rho Q.
112 \]
113 They satisfy exact completeness:
114 \[
115 \sum_aR_a^*R_a+R_{\mathrm{extra}}^*R_{\mathrm{extra}}
116 =\sum_aV_aV_a^*+Q=S+Q=I.
117 \]
118 For finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family $I_J\otimes R_a$, including the complement term. Positivity is not merely checked on a single matrix size.
119 
120 \subsection{Exact correction, including coherent errors}
121 For every operator supported on the code, $P\rho P=\rho$, the central identity is
122 \[
123 \boxed{\mathcal R(V_a\rho V_b^*)=\delta_{ab}\rho.}
124 \]
125 Consequently normalized mixtures satisfy
126 \[
127 \mathcal R\!\left(\sum_ap_aV_a\rho V_a^*\right)=\rho
128 \quad\text{when}\quad\sum_ap_a=1.
129 \]
130 The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error $E_z=\sum_a z_aV_a$, the calculation of the cross terms yields
131 \[
132 \mathcal R(E_z\rho E_z^*)=\left(\sum_a z_a\overline{z_a}\right)\rho.
133 \]
134 Thus the same map also corrects normalized coherent superpositions. This construction supplies the recovery stage used in the full Knill--Laflamme proof that follows.
135 
136 
137 

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