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 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) ] .

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 \(D_t\) 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 \(\psi \in \mathbb {C}^{D_0}\) 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 \(G_t\) 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) ] .

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 ] .