Dependencies

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

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

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

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

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

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

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

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

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,

The actual operator bounds are

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.

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

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

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

including zero or subnormalized \(v\).

Let \(A,C,W,X,Y,Z\) be finite index sets. Choose arbitrary fixed endpoint maps \(E:\mathbb {C}^A\to \mathbb {C}^W\), \(F:\mathbb {C}^A\to \mathbb {C}^X\), \(\widetilde E:\mathbb {C}^C\to \mathbb {C}^Y\), and \(\widetilde F:\mathbb {C}^C\to \mathbb {C}^Z\). For nonnegative real weights \(\lambda ,\mu \), define

The second ambient vector uses the weights \(\mu \). For the coordinate sampled factors of Definition 15.1.1, put \(U_j^{\mathrm{amb}}=EU_j\widetilde E^\dagger \) and \(V_j^{\mathrm{amb}}=FV_j\widetilde F^\dagger \). Then

In particular, the represented ambient vector is the endpoint transport of its paired coordinate Schmidt vector. These are the ambient-coordinate forms of [ Ope26a , Equation (5.6) ] .

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

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 \(\overline C_t=\overline G_{t-1}\cdots \overline G_0\),

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,

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

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) ] .

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

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

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

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

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) ] .

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

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

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) ] .

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

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

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.

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

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

Then

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.

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

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

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

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

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

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

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

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

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

Let \(S\) be finite. At each slot \(s\), let \(A_s,C_s\) be finite endpoint supports with nonnegative weights \(\lambda _s,\mu _s\). Take independent standard circular complex Gaussian coordinates \(g^{(j,s)}_{ac}\), with \(\mathbb E|g^{(j,s)}_{ac}|^2=1\), and independent Gaussian factors across slots. For \(k{\gt}0\), define

The product law is a probability measure, and both \(z_{il}\) and every conjugate pair product are integrable. Set \(p_i=\prod _s\lambda _s(i_s^1)\lambda _s(i_s^2)\) and \(\widetilde p_l=\prod _s\mu _s(l_s^1)\mu _s(l_s^2)\). Their covariance is (5). If each endpoint weight list sums to one, both product weight lists also sum to one. For nonempty \(S\), \(\mathbb E z_{il}=0\). These are the actual independent coefficients in [ Ope26a , Equations (5.7), (5.8), and (5.11) ] .

For the actual circular complex Gaussian law, each ambient factor \(U_j^{\mathrm{amb}},V_j^{\mathrm{amb}}\) is matrix-valued integrable; this integrability holds for arbitrary fixed real coefficient weights. If \(k{\gt}0\) and the endpoint weights are nonnegative, both \(\widehat X_{\mathrm{amb}}\) and \(\Delta _{\mathrm{amb}}\) are integrable, entrywise and as matrices, and

These expectation identities hold for arbitrary fixed endpoint maps; the isometry equations govern the separate norm-preservation estimate of Theorem 15.2.2. This is the unbiased replacement in [ Ope26a , Equation (5.6) ] .

Let \(G_t\) be contractions on the prescribed changing memory spaces, and let \(\psi ,K\) satisfy the hypotheses of Theorem 17.3.1. Choose a finite marked set \(S\) and \(\delta \geq 0\), with \(\lVert A_t-G_t\rVert \leq \delta \) for every \(t\in S\). Set

For the actual densities of (15),

If \(|S|{\gt}0\) and \(\epsilon \geq 0\), the choice \(\delta =\epsilon /(8|S|)\) gives error at most \(\epsilon /2\). For \(S=\varnothing \), \(H=G\) and the density error is exactly zero, for arbitrary input, readout, and rescaling parameter. Unmarked private stages remain exact. If all stages are approximated with a uniform error bound and \(H_t=A_t/(1+\delta )\) for every \(t\), the corresponding estimate is \(4n\delta \); for \(n{\gt}0\) and \(\epsilon \geq 0\), taking \(\delta =\epsilon /(8n)\) gives the same half-accuracy bound. With the nonprivate gate count \(M=|S|\), this is [ Ope26a , Equation (5.3) ] .

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 \(c_\beta \),

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) ] .

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 \(\kappa \geq 0\),

Define

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

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