Dependencies

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) ] .

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) ] .

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) ] .

Let \(J\) be finite, let \(\mu \) be a measure, and let \(z_j\) be complex functions with every product \(z_i\overline{z_j}\) integrable. Suppose \(\int z_i\overline{z_j}\, d\mu =\delta _{ij}v_i\), where \(v_i\in \mathbb {R}\). For complex numbers \(a_j\) and arbitrary rectangular matrices \(A_j\),

Both squared integrands are integrable. If \(\mu \) is a probability measure, then

These are the moment calculation and expectation estimate used in [ Ope26a , Equations (5.16) and (5.18) ] .

For nonnegative diagonal probability matrices \(\tau ,\widetilde\tau \) and an arbitrary rectangular matrix \(W\) of compatible size,

More generally, let \(R,\widetilde R\) be arbitrary positive semidefinite trace-one matrices and let \(O\) be a rectangular contraction between their spaces. Then

Zero eigenvalues and different input dimensions are allowed. These are the comparisons in [ Ope26a , Equations (5.15) and (5.17) ] .

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 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) ] .

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

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

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

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

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) ] .

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) ] .