Typical-projector bounds for finite quantum sources and cq ensembles

Diagonalize $\rho ={\sum }_{j:{\lambda }_j>0}{\lambda }_j\mathrm{\mid }j⟩⟨j\mathrm{\mid }$ by [F4], and sample $J$ with law $\mathbb{P}(J=j)={\lambda }_j$. The surprisal $-\ln {\lambda }_J$ is a finite-alphabet random variable of mean $S(\rho )$ by [F2]. For a product eigenvector with coordinates $j_1,\dots ,j_n$, its eigenvalue is ${\prod }_i{\lambda }_{j_i}$ and its surprisal is $-\ln ({\prod }_i{\lambda }_{j_i})={\sum }_i-\ln {\lambda }_{j_i}$ by [F7]. Let ${\mathrm{\Pi }}_n$ project onto product eigenvectors whose total surprisal lies in $[n(S(\rho )-\delta ),n(S(\rho )+\delta )]$. The probability of this set tends to one by [F1]. Every selected product eigenvalue is at most $e^{-n(S(\rho )-\delta )}$ and at least $e^{-n(S(\rho )+\delta )}$; the first bound gives the operator inequality, and the second together with total mass at most one gives the rank bound.

For an input word $x^n$, let $N_x$ count its occurrences of $x$. Define $T_n$ by $N_x=0$ when $p_x=0$ and $\mathrm{\mid }{N}_x\mathrm{/}n-p_x\mathrm{\mid }\le \kappa$ for every $p_x>0$. The coordinate indicators are independent under $p^{\otimes n}$ and have variance at most $1\mathrm{/}4$; [F6]–[F9] give $p^{\otimes n}(T_n^c)\le \mathrm{\mid }\mathcal{X}\mathrm{\mid }\mathrm{/}(4n{\kappa }^2)\to 0$. For each $x$ with $p_x>0$, $\bar{\sigma }⪰p_x{\sigma }_x$ by [F3] and [F10]; a vector in $\ker \bar{\sigma }$ therefore has zero quadratic form for ${\sigma }_x$, so [F5] gives $\mathrm{supp}{\sigma }_x\subseteq \mathrm{supp}\bar{\sigma }$. The spectral logarithm on the latter positive support is bounded in finite dimension by [F4], making ${\mu }_x:=\mathrm{tr}({\sigma }_x[-\ln \bar{\sigma }])$ finite for every such $x$. Define this family only for $p_x>0$; the zero-probability symbols never occur in $T_n$. Choose $\kappa >0$ small enough that the empirical averages of each of the two finite families $S({\sigma }_x)$ and ${\mu }_x$, restricted to $p_x>0$, differ from their $p$-averages by at most $\delta \mathrm{/}2$ on $T_n$. These choices are possible because both restricted families are finite and their entries are finite.

For each $x$, diagonalize ${\sigma }_x$ and, conditional on a fixed $x^n\in T_n$, measure each tensor factor in its chosen eigenbasis. The resulting surprisal variables $Z_i=-\ln {\lambda }_{J_i\mathrm{\mid }{x}_i}$ are independent, finite, and uniformly bounded over the finite family of positive eigenvalues. By [F7], the product eigenvalue surprisal is their sum. Their means sum to ${\sum }_{i}S({\sigma }_{x_i})$, which is within $n\delta \mathrm{/}2$ of $nH(B\mathrm{\mid }X)$ by step 1.2. Their sum has variance at most $n{L}^2$ for a fixed finite bound $L$; independence makes cross-covariances zero. Chebyshev [F6] shows that the probability of leaving the window $[n(H(B\mathrm{\mid }X)-\delta ),n(H(B\mathrm{\mid }X)+\delta )]$ tends to zero uniformly over $x^n\in T_n$. Project onto the product eigenvectors whose surprisal lies in this window. Each selected eigenvalue is at most $e^{-n(H(B\mathrm{\mid }X)-\delta )}$ and at least $e^{-n(H(B\mathrm{\mid }X)+\delta )}$, proving conditional acceptance, the operator bound, and the rank bound as in step 1.1.

By [F10], $\bar{\sigma }$ is a state and $\bar{\sigma }-p_x{\sigma }_x={\sum }_{z\mathrm{\ne }x}{p}_z{\sigma }_z⪰0$. Any $v\in \ker \bar{\sigma }$ has $⟨{\sigma }_{x}v,v⟩=0$ when $p_x>0$, so [F5] gives ${\sigma }_{x}v=0$. Hence $\mathrm{supp}{\sigma }_x\subseteq \mathrm{supp}\bar{\sigma }$. On that support, the variable $-\ln {\lambda }_J$ obtained by measuring ${\sigma }_x$ in an eigenbasis of $\bar{\sigma }$ is finite and uniformly bounded over $x$ with $p_x>0$ and positive eigenvalue index. Its mean is ${\mu }_x$. The $p$-average of these means is $\mathrm{tr}(\bar{\sigma }[-\ln \bar{\sigma }])=S(\bar{\sigma })$ by [F2], [F3], and [F8].

For fixed $x^n\in T_n$, measuring the product state ${⨂}_i{\sigma }_{x_i}$ in the product eigenbasis of ${\bar{\sigma }}^{\otimes n}$ gives independent surprisal variables with means ${\mu }_{x_i}$ and a common finite bound on their absolute values. By [F7], each product eigenvalue surprisal is the sum of the coordinate surprisals. Their means sum to within $n\delta \mathrm{/}2$ of $nS(\bar{\sigma })$ by step 1.2, and their sum has variance at most $n{L}^{\mathrm{\prime }2}$ for a finite $L^{\mathrm{\prime }}$. Chebyshev [F6] therefore bounds uniformly by a quantity tending to zero the probability of leaving the $\delta$-window around $nS(\bar{\sigma })$. The spectral projector of ${\bar{\sigma }}^{\otimes n}$ for that window accepts every such conditional product state with probability at least $1-u_n$. Its rank and compressed-operator bounds follow from its eigenvalue window exactly as in step 1.1.

Take $u_n$ to be the maximum of the uniform failure bounds in steps 2.1 and 3.1; both tend to zero, so $u_n\to 0$. The global state acceptance follows from step 1.1 applied to $\bar{\sigma }$; all conditional and operator/rank assertions are those proved in steps 2.1 and 3.1. The constructions use only finite spectra, finite sums, and finite probability spaces, so no choice axiom is invoked.