15 Expected Trace Error of Corrected Sources

The Gaussian replacement of a Schmidt source is unbiased. Its centered entries have diagonal covariance, and fresh independent slots preserve that covariance under tensor products. Quarter-power comparisons and finite block aggregation then bound the expected trace error for a fixed contraction between the complete free input spaces. Finite branch sums and insertion of physical matrix units preserve the required trace-norm estimates. These are the source calculations in [ Ope26a , Theorem 5.2 ] .

15.1 An unbiased product-operator source

Definition 15.1.1 Sampled Schmidt-source operators

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Let \(A,C\) be finite sets with nonnegative weights \(\lambda _a,\mu _c\), and let \(k\) be a positive integer. In the paired coordinate bases, put \(|\eta \rangle =\sum _a\sqrt{\lambda _a}|a,a\rangle \) and \(|\widetilde\eta \rangle =\sum _c\sqrt{\mu _c}|c,c\rangle \). For independent standard circular complex Gaussian coordinates \(g^{(j)}_{ac}\), with \(\mathbb E|g^{(j)}_{ac}|^2=1\), set

\begin{align} (U_j)_{ac} & =(\lambda _a\mu _c)^{1/4}g^{(j)}_{ac}, \label{eq:source_u} \\ (V_j)_{bd} & =(\lambda _b\mu _d)^{1/4}\overline{g^{(j)}_{bd}}, \label{eq:source_v} \\ \widehat X & =\frac1k\sum _{j=0}^{k-1}U_j\otimes V_j, \label{eq:source_sampled} \\ X & =|\eta \rangle \langle \widetilde\eta |, \label{eq:source_target} \\ \Delta & =\widehat X-X. \label{eq:source_correction} \end{align}

Thus \(X_{(a,b),(c,d)}=\sqrt{\lambda _a\mu _c}\delta _{ab}\delta _{cd}\), and

\begin{align} \widehat X_{(a,b),(c,d)} & =(\lambda _a\lambda _b\mu _c\mu _d)^{1/4} \frac1k\sum _{j=0}^{k-1}g^{(j)}_{ac}\overline{g^{(j)}_{bd}}. \label{eq:source_sampled_entry} \end{align}

This is [ Ope26a , Equation (5.6) ] .

Theorem 15.1.2 Unbiasedness and actual correction-entry covariance

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For the Gaussian probability law of Definition 15.1.1, \(\widehat X\) and \(\Delta \) are integrable, with \(\mathbb E\widehat X=X\) and \(\mathbb E\Delta =0\). The correction entries are precisely

\begin{align} \Delta _{(a,b),(c,d)} & =(\lambda _a\lambda _b\mu _c\mu _d)^{1/4} \left(\frac1k\sum _{j=0}^{k-1}g^{(j)}_{ac} \overline{g^{(j)}_{bd}}-\delta _{ab}\delta _{cd}\right). \label{eq:source_correction_entry} \end{align}

Every conjugate pair product is integrable. For \(p,r\in A\times A\) and \(q,t\in C\times C\),

\begin{align} \mathbb E[\Delta _{pq}\overline{\Delta _{rt}}] & =\frac1k\sqrt{\lambda _{p_1}\lambda _{p_2}\mu _{q_1}\mu _{q_2}} \delta _{pr}\delta _{qt}. \label{eq:source_entry_covariance} \end{align}

No normalization of the weight lists is required for these identities. These are the unbiasedness and covariance calculations in [ Ope26a , Equation (5.6) and Equations (5.7)–(5.8) ] .

Proof ▶

Expanding the Kronecker product gives (6). The quarter weight times \(\delta _{ab}\delta _{cd}\) equals \(X\) entrywise, giving (7). The average of the centered Gaussian products has mean zero and covariance \(k^{-1}\) times the product of coordinate deltas, proving (8). Its \(L^2\) bound gives entry and pair-product integrability. Finite coordinate evaluation gives matrix integrability and permits the entrywise expectations to be combined into the matrix identities.

15.2 Finite branch sums and physical matrix entries

Theorem 15.2.1 Complex coefficients in finite trace-norm sums

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For arbitrary rectangular complex matrices of a fixed size, the trace norm satisfies \(\lVert 0\rVert _1=0\), \(\lVert A+B\rVert _1\leq \lVert A\rVert _1+\lVert B\rVert _1\), \(\lVert cA\rVert _1=|c|\lVert A\rVert _1\), and \(\lVert A^\dagger \rVert _1=\lVert A\rVert _1\). For a finite set \(B\) of branches and complex coefficients \(c_\beta \),

\begin{align} \Bigl\lVert \sum _{\beta \in B}c_\beta A_\beta \Bigr\rVert _1 & \leq \sum _{\beta \in B}|c_\beta |\lVert A_\beta \rVert _1. \label{eq:source_branch_sum} \end{align}

Taking all \(c_\beta =1\) gives the unweighted finite-sum estimate; the empty branch family gives zero. These are the trace-norm properties used in the summation leading to [ Ope26a , Equation (5.19) ] .

Proof ▶

For a contraction \(V\), expand the trace pairing of the finite sum into \(\sum _\beta \overline{c_\beta }\operatorname{tr}(A_\beta ^\dagger V)\). The triangle inequality and trace-norm duality give (9), including the additive bound and zero case. An attaining contraction for \(A\) gives the reverse scalar inequality, hence exact homogeneity. Adjointing the test contraction and cyclically permuting the trace gives \(\lVert A^\dagger \rVert _1\leq \lVert A\rVert _1\); applying this to the adjoint gives equality.

Theorem 15.2.2 Exact physical matrix-unit reinsertion

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Let \(A\) be any rectangular complex matrix, and let \(K,L\) be isometric embeddings of its ket and bra coordinate spaces. Thus \(K^\dagger K=\mathbb {1}\) and \(L^\dagger L=\mathbb {1}\); both embeddings are operator contractions, and

\begin{align} \lVert KAL^\dagger \rVert _1 & =\lVert A\rVert _1. \label{eq:source_isometric_insertion} \end{align}

In particular, for finite physical ket and bra registers with coordinates \(x,y\), let \(J_xv=|x\rangle \otimes v\) and \(J_yw=|y\rangle \otimes w\). These maps are isometries and satisfy

\begin{align} J_xAJ_y^\dagger & =|x\rangle \langle y|\otimes A, \label{eq:source_matrix_unit} \\ \lVert |x\rangle \langle y|\otimes A\rVert _1 & =\lVert A\rVert _1. \label{eq:source_physical_reinsertion} \end{align}

The physical registers may differ, and no positivity or Hermiticity assumption on \(A\) is required. This is the reinsertion step following [ Ope26a , Equation (5.18) ] .

Proof ▶

The isometry equations imply the operator contraction bounds. Applying left and right trace-norm contraction gives one direction of (10). For the reverse direction, apply the adjoint contractions to \(KAL^\dagger \) and use \(K^\dagger (KAL^\dagger )L=A\). The identities \(J_x^\dagger J_x=\mathbb {1}\) and (11) follow by coordinate evaluation. Substitute these isometries into (10) to obtain (12).

15.3 Covariance and weighted block errors

Theorem 15.3.1 Expected trace error from diagonal covariance

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Let \(I,L,T,U\) be finite index sets, let \(p_i,\widetilde p_l\) be nonnegative probability lists, and let \(\tau ,\widetilde\tau \) be nonnegative diagonal trace-one matrices on the crossing-support spaces \(T,U\). Let \(O:\mathbb {C}^{I\times T}\to \mathbb {C}^{L\times U}\) be a contraction, with blocks \(O_{li}:\mathbb {C}^T\to \mathbb {C}^U\). On a probability space, let the complex coefficients \(z_{il}\) have integrable conjugate pair products and suppose, for \(\kappa \geq 0\),

\begin{align} \mathbb E[z_{il}\overline{z_{i'l'}}] & =\kappa \sqrt{p_i\widetilde p_l}\delta _{ii'}\delta _{ll'}. \label{eq:source_covariance} \end{align}

Define

\begin{align} Z & =\sum _{i,l}z_{il}\tau ^{1/2}O_{li}^{T}\widetilde\tau ^{1/2}, \label{eq:source_exterior_input} \\ H & =\sum _{i,l}z_{il}\widetilde\tau ^{1/4}O_{li}\tau ^{1/4}. \label{eq:source_quarter_sum} \end{align}

The transpose is ordinary full transpose, preserving the coefficients. The functions \(\lVert Z\rVert _1\) and \(\lVert H\rVert _2\) are integrable, and

\begin{align} \mathbb E\lVert H\rVert _2^2 & =\kappa \sum _{i,l}\sqrt{p_i\widetilde p_l} \lVert \widetilde\tau ^{1/4}O_{li}\tau ^{1/4}\rVert _2^2, \label{eq:source_block_moment} \\ \mathbb E\lVert Z\rVert _1 & \leq \sqrt\kappa . \label{eq:source_expected_error} \end{align}

These are the covariance implication and block calculation in [ Ope26a , Equations (5.14)–(5.18) ] .

Proof ▶

Quarter-power comparison and ordinary transpose give \(\lVert Z\rVert _1\leq \lVert H\rVert _2\). Diagonal covariance and Theorem 14.3.1 give (16). Put \(R=\operatorname{diag}(p)\otimes \tau \) and \(\widetilde R=\operatorname{diag}(\widetilde p)\otimes \widetilde\tau \). Their diagonal lists \(p_i\tau _t\) and \(\widetilde p_l\widetilde\tau _u\) sum to one. Entrywise expansion gives the exact block aggregation

\begin{align} \sum _{i,l}\sqrt{p_i\widetilde p_l} \lVert \widetilde\tau ^{1/4}O_{li}\tau ^{1/4}\rVert _2^2 & =\lVert \widetilde R^{1/4}OR^{1/4}\rVert _2^2\leq 1. \label{eq:source_aggregation} \end{align}

Pair-product integrability makes the Gram matrix \(Z^\dagger Z\) almost everywhere measurable; taking its positive square root and trace gives almost everywhere strong measurability of \(\lVert Z\rVert _1\). The integrable square of \(\lVert H\rVert _2\) gives its integrability, and the pointwise comparison gives integrability of \(\lVert Z\rVert _1\). Finally, Cauchy–Schwarz on the probability space yields \(\mathbb E\lVert Z\rVert _1\leq \mathbb E\lVert H\rVert _2 \leq \sqrt{\mathbb E\lVert H\rVert _2^2}\leq \sqrt\kappa \).

Theorem 15.3.2 Trace error under the constructed independent Gaussian law

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Let \(S\) be a finite set of corrected slots, with \(s=|S|\). At each slot, choose finite ket and bra supports \(A_s,C_s\) and normalized nonnegative endpoint weights. The corrected-register index sets are \(I=\prod _s(A_s\times A_s)\) and \(L=\prod _s(C_s\times C_s)\). Construct the Gaussian coefficients and their independent product law as in Theorem 14.1.4. Let \(\tau ,\widetilde\tau \) be nonnegative diagonal trace-one matrices, and let \(O\) be a contraction between the complete free registers. For every positive sample count \(k\), define \(Z,H\) by (14) and (15). Then \(\lVert Z\rVert _1\) is integrable, the second moment of \(H\) is (16) with \(\kappa =k^{-s}\), and

\begin{align} \mathbb E\lVert Z\rVert _1 & \leq k^{-s/2}. \label{eq:source_gaussian_error} \end{align}

Here the expectation uses the constructed independent Gaussian law. This is [ Ope26a , Equation (5.18) ] .

Proof ▶

The actual independent Gaussian law supplies the diagonal covariance (13) with \(\kappa =k^{-s}\) and supplies pair-product integrability. Normalization of the endpoint lists gives normalization of the product lists \(p,\widetilde p\). Apply Theorem 15.3.1 and use \(\sqrt{k^{-s}}=k^{-s/2}\) for \(k{\gt}0\).