Dependencies

Under the preceding hypotheses,

These two inequalities remain true without the upper bounds on \(\lVert \Psi \rVert \) and \(\lVert \Xi \rVert \); the second is then a signed trace-deficit bound. If an arbitrary operator \(\sigma \) satisfies \(\lVert \sigma -\rho _\Psi \rVert _1\leq \kappa \), then, with the original upper norm bounds,

Positivity and Hermiticity of \(\sigma \) are unnecessary. This is the error accumulation in [ Ope26a , Section 8.1 ] .

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 \(A,C\) 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

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

This is [ Ope26a , Equation (5.6) ] .

Independent bijections of the finite row and column coordinates of a complex matrix preserve its nuclear norm, including when either coordinate set is empty.

Let \(\mathcal H_A,\mathcal H_E\) be finite-dimensional Euclidean spaces, \(\Omega \in \mathcal H_A\) a unit vector, and \(\Xi \in \mathcal H_E\) with \(\lVert \Xi \rVert \leq 1\). For \(\Psi \in \mathcal H_A\otimes \mathcal H_E\) with \(\lVert \Psi \rVert \leq 1\), write \(\rho _\Psi =\operatorname{Tr}_E|\Psi \rangle \langle \Psi |\). Then

In particular, if \(\lVert \Psi -\Omega \otimes \Xi \rVert \leq \varepsilon \) and \(1-\delta \leq \lVert \Xi \rVert \), the density error is at most \(2\varepsilon +2\delta \). Neither \(\Xi \ne 0\) nor \(\delta \leq 1\) is required. These estimates give the explicit constants in the passage from the product reference to the physical density in [ Ope26a , Section 8.1 ] .

Let \(Z\) be an arbitrary rectangular matrix. Let \(K\) and \(\widetilde K\) be contractions from its respective output and input spaces into a common physical space tensored with a discard space. Then

The same norm bound holds for left and right contractions alone, and partial trace alone cannot increase the trace norm. This is [ Ope26a , Equation (5.13) ] .

For a complex rectangular matrix \(A\), let \(\lVert A\rVert _1\) be the sum of its singular values. Equivalently, \(\lVert A\rVert _1=\sum _j\sqrt{\lambda _j(A^\dagger A)}\); for square matrices this is the trace norm. Every rectangular contraction \(U\) of the same size satisfies \(|\operatorname{tr}(A^\dagger U)|\leq \lVert A\rVert _1\), and some such \(U\) satisfies \(\operatorname{tr}(A^\dagger U)=\lVert A\rVert _1\). This is the trace-norm duality used in [ Ope26a , Equation (5.13) ] .

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 \(D_t\) be finite index sets, and let \(G_t,H_t:\mathbb {C}^{D_t}\to \mathbb {C}^{D_{t+1}}\) be contractions for every \(t\in \mathbb {N}\). Define the chronological prefixes by \(\Pi _G(0)=\mathbb {1}\) and \(\Pi _G(t+1)=G_t\Pi _G(t)\). Each prefix is a contraction, including the identity on an empty space, and

A uniform error bound \(\lVert G_t-H_t\rVert \leq \delta \) gives the bound \(n\delta \). More generally, for a finite marked set \(S\subset \mathbb {N}\) and \(\delta \geq 0\), errors at most \(\delta \) on \(S\) and exact equality outside \(S\) give \(\lVert \Pi _G(n)-\Pi _H(n)\rVert \leq |S|\delta \). In a real normed space, if \(\lVert G\rVert \leq 1\) and \(\lVert A-G\rVert \leq \delta \) with \(\delta \geq 0\), then \(\lVert A/(1+\delta )\rVert \leq 1\) and \(\lVert A/(1+\delta )-G\rVert \leq 2\delta \). These are the contraction estimates used before [ Ope26a , Equation (5.3) ] .

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 finite Euclidean spaces, the rectangular outer product has entries \((|u\rangle \langle v|)_{ij}=u_i\overline{v_j}\) and satisfies \(\lVert |u\rangle \langle v|\rVert _1\leq \lVert u\rVert \lVert v\rVert \). For vectors \(v,w\) in the same space,

Discarding a finite register preserves this upper bound. In particular, if \(\lVert v\rVert ,\lVert w\rVert \leq 1\), both the full and retained density errors are at most \(2\lVert v-w\rVert \). Zero vectors and zero-dimensional spaces are allowed. This is the vector-to-density estimate preceding [ Ope26a , Equation (5.3) ] .

Let \(\psi \in \mathbb {C}^{D_0}\) satisfy \(\lVert \psi \rVert \leq 1\), and let \(K:\mathbb {C}^{D_n}\to \mathbb {C}^{P\times E}\) be a common contraction into finite physical and discarded registers. Define the actual final vector and retained density by

For arbitrary gate families, \(\lVert v_H-v_G\rVert \leq \lVert \Pi _H(n)-\Pi _G(n)\rVert \). If both families are contractions, \(v_G,v_H\) have norm at most one and

For \(n=0\) the densities coincide and their trace-norm difference is zero; this equality holds for arbitrary \(\psi ,K,G,H\). These are the actual vector and density estimates used in [ Ope26a , Equation (5.3) ] .

Let \(A,C\) be finite index sets and let \(W,X,Y,Z\) be arbitrary index sets. For endpoint matrices \(E,F,\widetilde E,\widetilde F\) of the indicated sizes, define the coordinate arrays \(u_{ab}(w,x)=E_{wa}F_{xb}\) and \(v_{cd}(y,z)=\widetilde E_{yc}\widetilde F_{zd}\). Every matrix \(M\) with rows indexed by \(A\times A\) and columns indexed by \(C\times C\) satisfies

No orthonormality or spanning condition on the endpoint matrices is needed.

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) ] .

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

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

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) ] .