18 Source Frames, Global Sampling, and Idle Registers

Schmidt decomposition gives the actual local frames needed for Gaussian source replacement. A single product probability space samples every position and branch label, while multilinearity expands errors by corrected positions. Chronologically appended normalized pure registers can be kept idle and discarded without changing the original physical density. These are the constructions used in [ Ope26a , Theorem 5.2 ] . Applying them to a distributed circuit requires its chronological memory spaces, participating parties, source maps, and physical readout. The weighted error estimate also requires a contraction bound for the exterior operator.

18.1 Actual Schmidt source frames

Theorem 18.1.1 Schmidt frames and an unbiased replacement

✓

Let \(L,R\) be finite index sets and let \(\psi \in \mathbb {C}^L\otimes \mathbb {C}^R\). Set \(r=\min (|L|,|R|)\). There are nonnegative weights \(\lambda _a\) and column isometries \(E:\mathbb {C}^r\to \mathbb {C}^L\), \(F:\mathbb {C}^r\to \mathbb {C}^R\) such that

\begin{align} \psi & =\sum _{a=1}^r\sqrt{\lambda _a}\, E|a\rangle \otimes F|a\rangle , \label{eq:construction_schmidt_vector} \\ E^\dagger E & =F^\dagger F=I_r, \qquad \sum _a\lambda _a=\lVert \psi \rVert ^2. \label{eq:construction_schmidt_mass} \end{align}

Thus unit vectors give probability weight lists. The zero vector and zero-dimensional spaces are included.

For an arbitrary second vector \(\phi \in \mathbb {C}^{\widetilde L}\otimes \mathbb {C}^{\widetilde R}\), choose its corresponding frames \(\widetilde E,\widetilde F\) and weights \(\mu \). For every positive sample count \(k\), the ambient product replacement of Definition 17.1.1 is integrable and

\begin{align} X_{\mathrm{amb}} & =|\psi \rangle \langle \phi |, \qquad \mathbb E\widehat X_{\mathrm{amb}}=|\psi \rangle \langle \phi |. \label{eq:construction_schmidt_replacement} \end{align}

The ket and bra dimensions may differ. This supplies the Schmidt coordinates in [ Ope26a , Equation (5.6) ] .

Proof ▶

Schmidt decomposition gives orthonormal families and nonnegative weights with mass \(\lVert \psi \rVert ^2\) [ Wol12 , Proposition 1.1 ] . Reindex the families into the ambient spaces. Their Gram matrices are identities, giving the column-isometry equations. Apply the same argument to \(\phi \). Their outer product is exactly \(X_{\mathrm{amb}}\), so integrability and unbiasedness follow from Theorem 17.1.3.

18.2 Common private source sectors

Definition 18.2.1 Adjoint extension of a local domain

✓

For finite index sets \(N,M,Q\), let \(B:\mathbb {C}^N\to \mathbb {C}^M\) be rectangular and let \(J:\mathbb {C}^N\to \mathbb {C}^Q\) satisfy \(J^\dagger J=I_N\). Its adjoint domain extension is \(\overline B=BJ^\dagger \). It satisfies

\begin{align} \overline BJ & =B, \qquad \overline BJJ^\dagger =\overline B, \label{eq:construction_domain_recovery} \\ \lVert \overline B\rVert & =\lVert B\rVert , \qquad \lVert J\psi \rVert =\lVert \psi \rVert . \label{eq:construction_domain_norm} \end{align}

Thus local contractions remain contractions on their common ambient domain. These equalities allow empty domains, empty codomains, and zero or subnormalized vectors.

Proof ▶

Multiplication by \(J^\dagger J=I_N\) proves recovery and range support. The isometry and its adjoint have operator norm at most one. Hence \(\lVert BJ^\dagger \rVert \leq \lVert B\rVert \) and \(\lVert B\rVert =\lVert BJ^\dagger J\rVert \leq \lVert BJ^\dagger \rVert \). Apply the same contraction and left-inverse argument to vectors.

Theorem 18.2.2 Canonical private branch sectors

✓

Let \(\Xi \) be finite and let \(N_\xi \) be arbitrary finite branch index sets. Write \(Q=\bigoplus _{\xi \in \Xi }\mathbb {C}^{N_\xi }\) and let \(J_\xi :\mathbb {C}^{N_\xi }\to Q\) be the canonical sector inclusions. For arbitrary rectangular \(B_\xi \) with domain \(\mathbb {C}^{N_\xi }\), set \(\overline B_\xi =B_\xi J_\xi ^\dagger \). Then

\begin{align} \overline B_\xi J_\xi & =B_\xi , \qquad \overline B_\xi J_\zeta =0\quad (\xi \ne \zeta ), \label{eq:construction_branch_recovery} \\ \lVert \overline B_\xi \rVert & =\lVert B_\xi \rVert , \qquad \lVert J_\xi \psi \rVert =\lVert \psi \rVert . \label{eq:construction_branch_norm} \end{align}

For any finite external memory \(R\), extension of a map with domain \(\mathbb {C}^{N_\xi }\otimes \mathbb {C}^R\) through \(J_\xi \otimes I_R\) likewise preserves its exact operator norm. For a second family of finite private sectors \(L_\xi \), their inclusions \(\widetilde J_\xi \) satisfy

\begin{align} \lVert (J_\xi \otimes \widetilde J_\xi )\psi \rVert & =\lVert \psi \rVert . \label{eq:construction_bipartite_sector_norm} \end{align}

No bound on the private branch dimensions is required. These are the common private slot spaces in [ Ope26a , Theorem 5.2 ] .

Proof ▶

Canonical sector inclusions satisfy \(J_\xi ^\dagger J_\xi =I\) and \(J_\xi ^\dagger J_\zeta =0\) for distinct branches. Apply Definition 18.2.1. Tensor products of these isometries, including an identity on external memory, remain isometries and give the two tensor norm-preservation statements.

18.3 Multilinear contraction and correction positions

Definition 18.3.1 Finite coefficient contraction

✓

Let \(P\) be finite. For each position \(p\), let \(R_p,C_p\) be finite row and column sets. For every label \(x\in \prod _{p\in P}(R_p\times C_p)\), fix a matrix \(M_x\) with the same output row and column sets. Define the multilinear contraction

\begin{align} \mathcal C(X) & =\sum _x \left(\prod _{p\in P}(X_p)_{x_p^1x_p^2}\right)M_x, \label{eq:construction_contraction} \\ \mathcal C(X)_{ab} & =\sum _x(M_x)_{ab} \prod _{p\in P}(X_p)_{x_p^1x_p^2}. \label{eq:construction_contraction_entry} \end{align}

Each \(X_p\) is an arbitrary rectangular matrix. The coefficient matrices are deterministic and unrestricted.

Theorem 18.3.2 Expansion over corrected positions

✓

Fix exact sources \(X_p\), corrections \(\Delta _p\), and a set \(T\subseteq P\). Let \(X^{T,+}_p=X_p+\Delta _p\) on \(T\) and \(X_p\) outside \(T\). For each \(S\subseteq T\), let \(Z^S_p=\Delta _p\) on \(S\) and \(X_p\) outside \(S\). Then

\begin{align} \mathcal C(X^{T,+}) & =\sum _{S\subseteq T}\mathcal C(Z^S), \label{eq:construction_position_sum} \\ \mathcal C(X^{T,+})-\mathcal C(X) & =\sum _{\varnothing \ne S\subseteq T}\mathcal C(Z^S). \label{eq:construction_nonempty_sum} \end{align}

Taking \(T=P\) gives the full replacement expansion. For finite output spaces, its actual rectangular nuclear error satisfies

\begin{align} \lVert \mathcal C(X+\Delta )-\mathcal C(X)\rVert _1 & \leq \sum _{\varnothing \ne S\subseteq P} \lVert \mathcal C(Z^S)\rVert _1. \label{eq:construction_position_norm} \end{align}

The subsets count positions independently of the number of branch labels. This is the algebraic expansion in [ Ope26a , Equation (5.9) ] .

Proof ▶

Expand each corrected factor using multilinearity. Each choice of the correction rather than the exact source determines one subset \(S\). The empty subset contributes \(\mathcal C(X)\). Subtract that term and apply the finite nuclear-norm triangle inequality.

18.4 A single probability space for all branch labels

Definition 18.4.1 Global samples and fixed coordinate choices

✓

Let \(P\) be finite, let \(B_p\) be its finite branch-label sets, and let \(A_{pb},C_{pb}\) be finite Schmidt supports. For each \(p,b\), let \(\nu _{pb}\) be the standard circular complex Gaussian law on the coordinates \(\{ 0,\ldots ,k-1\} \times A_{pb}\times C_{pb}\). The nested law

\begin{align} \nu & =\bigotimes _{p\in P}\bigotimes _{b\in B_p}\nu _{pb} \label{eq:construction_global_law} \end{align}

is a probability measure. For \(S\subseteq P\) and a deterministic choice \(\sigma _p\in B_p\) at each \(p\in S\), define \(\pi _{S,\sigma }(\omega )_p=\omega _{p,\sigma _p}\). Its actual pushforward is

\begin{align} (\pi _{S,\sigma })_*\nu & =\bigotimes _{p\in S}\nu _{p,\sigma _p}. \label{eq:construction_selected_law} \end{align}

The branch choice is fixed before sampling. This realizes the fresh samples used in [ Ope26a , Theorem 5.2 ] .

Proof ▶

Projection of a finite product probability measure onto selected positions preserves the corresponding product measure. At each selected position, projection onto the fixed branch coordinate preserves its Gaussian law. Compose these projections to obtain (15).

Theorem 18.4.2 Selected moments on the global law

✓

For a positive sample count \(k\), use the selected quarter-weight correction coefficients \(c_{p,\sigma _p}\) from (6). Define

\begin{align} z_{il}(\omega ) & =\prod _{p\in S} c_{p,\sigma _p}(i_p,l_p)(\omega _{p,\sigma _p}), \label{eq:construction_selected_coefficient} \\ q_i & =\prod _{p\in S}\lambda _p(i_p^1)\lambda _p(i_p^2), \qquad \widetilde q_l=\prod _{p\in S}\mu _p(l_p^1)\mu _p(l_p^2). \label{eq:construction_selected_weights} \end{align}

The weights are nonnegative. On the single global law, \(z_{il}\) and every conjugate pair product are integrable, and

\begin{align} \mathbb E_\nu [z_{il}\overline{z_{i'l'}}] & =k^{-|S|}\sqrt{q_i\widetilde q_l}\, \delta _{ii'}\delta _{ll'}. \label{eq:construction_selected_covariance} \end{align}

If \(S\) is nonempty, \(\mathbb E_\nu z_{il}=0\). The inverse-sample exponent counts corrected positions; unselected positions and branch labels do not contribute. This is [ Ope26a , Equation (5.11) ] on the global probability space.

Proof ▶

The selected coefficients are compositions of the independent-slot coefficients with \(\pi _{S,\sigma }\). Its measure-preserving identity transports their integrability, covariance, and centeredness to the global law. The selected index set has cardinality \(|S|\).

Theorem 18.4.3 Unbiasedness of the actual coefficient-contracted sampled output

✓

Let \(k{\gt}0\) and fix a deterministic branch choice \(\sigma _p\) at every position \(p\in P\). Choose nonnegative Schmidt weights \(\lambda _p,\mu _p\) on the corresponding finite supports. Let \(\widehat X_p(\omega _{p,\sigma _p})\) be the actual sampled source of Definition 15.1.1, with exact source \(X_p\) and correction \(\Delta _p=\widehat X_p-X_p\). For the finite coefficient contraction of Definition 18.3.1, with finite output row and column sets, define

\begin{align} \widehat Y(\omega ) & =\mathcal C((\widehat X_p(\omega _{p,\sigma _p}))_{p\in P}), \qquad Y=\mathcal C((X_p)_{p\in P}), \label{eq:construction_sampled_output} \\ Y_S(\omega ) & =\mathcal C(Z^S(\omega )), \label{eq:construction_corrected_output} \end{align}

where \(Z^S_p=\Delta _p(\omega _{p,\sigma _p})\) on \(S\) and \(X_p\) outside \(S\). These are actual matrix-valued contractions. For each entry label \(x=(i_p,l_p)_{p\in P}\), their literal source-entry product is

\begin{align} \prod _{p\in P}(Z^S_p)_{i_pl_p} & =\left(\prod _{p\notin S}(X_p)_{i_pl_p}\right) z_{x|S}(\omega ), \label{eq:construction_corrected_entry_factorization} \end{align}

with \(z_{x|S}\) the actual selected coefficient of (16). Substitution into (9) gives the corresponding finite matrix sum. In particular,

\begin{align} \widehat Y-Y & =\sum _{\varnothing \ne S\subseteq P}Y_S. \label{eq:construction_sampled_output_expansion} \end{align}

On the single global Gaussian law, every \(Y_S\) is integrable, \(\mathbb E_\nu Y_S=0\) for nonempty \(S\). The actual sampled output \(\widehat Y\) is integrable and

\begin{align} \mathbb E_\nu \widehat Y & =Y. \label{eq:construction_sampled_output_mean} \end{align}

The source weights need not sum to one, and the deterministic coefficient matrices have no norm restriction. This realizes the corrected-position expansion and its centered terms in [ Ope26a , Equations (5.6) and (5.9) ] .

Proof ▶

Each actual correction entry equals its selected Gaussian coefficient. Factor its product into the constant outside entries and the selected-position coefficient to obtain (21). Finite sums of these integrable scalar coefficients times fixed matrices prove integrability of \(Y_S\). Their nonempty selected products have zero mean by Theorem 18.4.2. Integrate the literal nonempty-position expansion. Adding the constant \(Y\) gives both integrability of \(\widehat Y\) and its exact matrix-valued expectation.

Theorem 18.4.4 Global-law expected error for a fixed branch choice

✓

Let \(k{\gt}0\). Assume each selected Schmidt list \(\lambda _p,\mu _p\) is nonnegative and sums to one. Let \(\tau ,\widetilde\tau \) be nonnegative diagonal trace-one matrices on finite spaces \(T,U\), and let \(O:\mathbb {C}^{I\times T}\to \mathbb {C}^{L\times U}\) be a contraction, where \(I=\prod _{p\in S}(A_{p,\sigma _p}\times A_{p,\sigma _p})\) and \(L=\prod _{p\in S}(C_{p,\sigma _p}\times C_{p,\sigma _p})\). For the actual selected coefficients of (16), set

\begin{align} Z_\sigma (\omega ) & =\sum _{i,l}z_{il}(\omega ) \tau ^{1/2}O_{li}^{T}\widetilde\tau ^{1/2}. \label{eq:construction_selected_error} \end{align}

The transpose is ordinary. The function \(\lVert Z_\sigma \rVert _1\) is integrable on \(\nu \), and

\begin{align} \mathbb E_\nu \lVert Z_\sigma \rVert _1 & \leq k^{-|S|/2}. \label{eq:construction_global_error} \end{align}

Zero weights and differing finite Schmidt supports are allowed. The empty selected set is included. This is the fixed-choice estimate of [ Ope26a , Equation (5.18) ] .

Proof ▶

Finite distributivity normalizes \(q,\widetilde q\). Apply Theorem 15.3.1 with \(\kappa =k^{-|S|}\), or transport the independent-slot estimate through \(\pi _{S,\sigma }\). This gives both integrability and (25) on the one global law.

18.5 Chronological fresh registers

Definition 18.5.1 Accumulated pure inventory

✓

Choose finite fresh registers \(B_t\) and pure vectors \(\gamma _t\in \mathbb {C}^{B_t}\). Define the finite chronological inventory \(J_0=\{ *\} \), \(J_{t+1}=J_t\times B_t\), with

\begin{align} \Gamma _0 & =1\in \mathbb {C}^{J_0}, \qquad \Gamma _{t+1}=\Gamma _t\otimes \gamma _t. \label{eq:construction_inventory_vector} \end{align}

Coordinate tensor products satisfy \((v\otimes w)_{ij}=v_iw_j\) and \(\lVert v\otimes w\rVert =\lVert v\rVert \lVert w\rVert \). Their actual outer products satisfy

\begin{align} |v\otimes w\rangle \langle v\otimes w| & =|v\rangle \langle v|\otimes |w\rangle \langle w|, \label{eq:construction_tensor_density} \\ \operatorname{tr}(|w\rangle \langle w|) & =\lVert w\rVert ^2. \label{eq:construction_pure_trace} \end{align}

Thus normalized \(\gamma _t\) give \(\lVert \Gamma _t\rVert =1\). Discarding any normalized pure factor \(w\) gives

\begin{align} \operatorname{Tr}_w(|v\otimes w\rangle \langle v\otimes w|) & =|v\rangle \langle v|, \label{eq:construction_pure_discard} \end{align}

including zero or subnormalized \(v\).

Definition 18.5.2 Appending a register while earlier registers remain idle

✓

For \(G:\mathbb {C}^D\to \mathbb {C}^{D'}\) and \(\gamma \in \mathbb {C}^B\), define \(A_{G,\gamma }:\mathbb {C}^D\to \mathbb {C}^{D'\times B}\) by \((A_{G,\gamma })_{(y,e),x}=G_{yx}\gamma _e\). For an arbitrary \(A:\mathbb {C}^D\to \mathbb {C}^{D'\times B}\) and an old inventory \(J\), its idle lift \(\mathcal L_J(A)\) is given by

\begin{align} \mathcal L_J(A)_{(y,(j,e)),(x,j')} & =A_{(y,e),x}\delta _{jj'}. \label{eq:construction_idle_coefficients} \end{align}

This is \(A\otimes I_J\) followed by the stated output permutation. On an arbitrary vector \(\xi \), \((\mathcal L_J(A)\xi )_{y,j,e}=\sum _x A_{(y,e),x}\xi _{x,j}\). In particular,

\begin{align} A_{G,\gamma }v & =Gv\otimes \gamma , \label{eq:construction_append_action} \\ \mathcal L_J(A_{G,\gamma })(v\otimes w) & =Gv\otimes (w\otimes \gamma ). \label{eq:construction_idle_action} \end{align}

The actual operator bounds are

\begin{align} \lVert A_{G,\gamma }\rVert & \leq \lVert G\rVert \lVert \gamma \rVert , \label{eq:construction_append_norm} \\ \lVert \mathcal L_J(A)\rVert & \leq \lVert A\rVert , \qquad \mathcal L_J(A-A’)=\mathcal L_J(A)-\mathcal L_J(A’), \label{eq:construction_idle_norm} \\ \lVert \mathcal L_J(A)-\mathcal L_J(A’)\rVert & \leq \lVert A-A’\rVert . \label{eq:construction_idle_error} \end{align}

Likewise \((K\otimes I_J)(v\otimes w)=Kv\otimes w\) and \(\lVert K\otimes I_J\rVert \leq \lVert K\rVert \) for arbitrary rectangular \(K\). No dimension lower bounds are needed.

Theorem 18.5.3 Actual chronological prefixes and their discarded densities

✓

Let \(D_t\) be finite changing memory spaces and let \(G_t:\mathbb {C}^{D_t}\to \mathbb {C}^{D_{t+1}}\). Define actual ideal and replacement stages on the accumulated inventory by

\begin{align} \overline G_t & =\mathcal L_{J_t}(A_{G_t,\gamma _t}), \qquad \overline A_t=\mathcal L_{J_t}(A_t), \label{eq:construction_chronological_gates} \end{align}

where \(A_t:\mathbb {C}^{D_t}\to \mathbb {C}^{D_{t+1}\times B_t}\) is arbitrary. With \(C_t=G_{t-1}\cdots G_0\) and \(\overline C_t=\overline G_{t-1}\cdots \overline G_0\),

\begin{align} \overline C_t(\psi \otimes \Gamma _0) & =C_t\psi \otimes \Gamma _t. \label{eq:construction_chronological_prefix} \end{align}

This equality requires no normalization assumptions. If every \(\gamma _t\) is normalized, discarding \(J_t\) gives exactly \(|C_t\psi \rangle \langle C_t\psi |\). Also,

\begin{align} \lVert \overline A_t-\overline G_t\rVert & \leq \lVert A_t-A_{G_t,\gamma _t}\rVert . \label{eq:construction_chronological_error} \end{align}

Contractive \(A_t\) give contractive \(\overline A_t\). Contractive \(G_t\) and normalized \(\gamma _t\) give contractive \(\overline G_t\). The replacement may correlate its working memory with the fresh output register.

For an arbitrary common readout \(K:\mathbb {C}^{D_t}\to \mathbb {C}^{P\times E}\), set \(\overline v_t=(K\otimes I_{J_t}) \overline C_t(\psi \otimes \Gamma _0)\). Then

\begin{align} \overline v_t & =KC_t\psi \otimes \Gamma _t, \label{eq:construction_chronological_readout} \\ \operatorname{Tr}_E\operatorname{Tr}_{J_t}(|\overline v_t\rangle \langle \overline v_t|) & =\operatorname{Tr}_E(|KC_t\psi \rangle \langle KC_t\psi |). \label{eq:construction_original_density} \end{align}

The density identity assumes only normalization of the supplied fresh vectors; the gates, readout, and input need not be normalized. It is the unchanged physical density preceding [ Ope26a , Equation (5.3) ] .

Proof ▶

Induct on the chronological prefix using (32). This gives (37). Normalization of \(\Gamma _t\) and (29) prove the discarded-density identity. The gate and error bounds follow from (33)–(35). The identity-amplified readout gives (39); discard \(J_t\) first and then \(E\) to obtain (40).

Theorem 18.5.4 Marked replacement error against the original physical density

✓

In the chronological construction, suppose \(\lVert \gamma _t\rVert =1\), \(\lVert G_t\rVert \leq 1\), \(\lVert \psi \rVert \leq 1\), and \(\lVert K\rVert \leq 1\). Choose a finite set of marked stages \(S\) and \(\delta \geq 0\), with \(\lVert A_t-A_{G_t,\gamma _t}\rVert \leq \delta \) for every \(t\in S\). Define the actual augmented replacement stages

\begin{align} H_t & =\begin{cases} \overline A_t/(1+\delta ), & t\in S, \\ \overline G_t, & t\notin S. \end{cases} \label{eq:construction_marked_inventory_gates} \end{align}

Put \(v_H=(K\otimes I_{J_n})H_{n-1}\cdots H_0 (\psi \otimes \Gamma _0)\) and define the actual physical densities

\begin{align} \rho _H & =\operatorname{Tr}_E\operatorname{Tr}_{J_n}(|v_H\rangle \langle v_H|), \qquad \rho _G=\operatorname{Tr}_E(|KC_n\psi \rangle \langle KC_n\psi |). \label{eq:construction_original_comparison} \end{align}

Then

\begin{align} \lVert \rho _H-\rho _G\rVert _1 & \leq 4|S|\delta . \label{eq:construction_original_error} \end{align}

Only marked stages require approximation. Unmarked stages are exact. Marks beyond the final stage merely weaken the bound. For \(\epsilon \geq 0\) and \(|S|{\gt}0\), taking \(\delta =\epsilon /(8|S|)\) gives error at most \(\epsilon /2\). This is the comparison with the original physical density in [ Ope26a , Equation (5.3) ] .

For \(S=\varnothing \), \(\rho _H=\rho _G\) and the nuclear error is exactly zero for arbitrary gates, replacements, readout, input, and \(\delta \), provided the supplied fresh vectors are normalized. For zero stages, the density identity holds for every augmented chain without normalization assumptions on future fresh vectors.

Proof ▶

Idle lifting bounds the augmented per-stage errors by the original marked errors, and the ideal augmented stages are contractions. The initial tensor vector has norm at most one, and identity amplification gives a contractive common readout. Theorem 17.3.2 bounds the augmented readout density error by \(4|S|\delta \). Discard the remaining owned register and use (40) to identify the ideal density with \(\rho _G\). Partial trace cannot increase the nuclear norm. The positive marked-count substitution gives the half-accuracy bound. An empty marked set leaves the ideal chain unchanged. With zero stages, the only inventory is its initial scalar vector.