Finite cq wiretap achievability with strong Holevo secrecy

By [F9], ${\chi }_E\ge 0$. Since $R<{\chi }_B-{\chi }_E$, choose $r$ with ${\chi }_E<r$ and $R+r<{\chi }_B$, and then choose $\eta >0$ so $r>{\chi }_E+2\eta$ and $R+r<{\chi }_B-2\eta$. Take $M_n=\lfloor e^{nR}\rfloor$, $L_n=\lceil e^{nr}\rceil$ and draw $M_n{L}_n$ independent words $X^n(m,l)$ with law $p^{\otimes n}$. For message $m$, the encoder inputs the mixed state ${\rho }_{m,n}=\frac{1}{L_n}{\sum }_{l=1}^{L_n}{⨂}_{i=1}^n{\rho }_{X_i(m,l)}\mathrm{.}$ This is a legitimate finite private randomization of the operational input state; the realized codebook is public.

For Bob use the global and conditional spectral projectors $P,Q_{x^n}$ from Exponential averaged spectral projectors for a finite cq ensemble, for the receiver ensemble. Put ${\mathrm{\Gamma }}_{m,l}=P{Q}_{X^n(m,l)}P$ and $G={\sum }_{m,l}{\mathrm{\Gamma }}_{m,l}$. Each ${\mathrm{\Gamma }}_{m,l}$ is nonnegative and ${\mathrm{\Gamma }}_{m,l}⪯P⪯I$. Using [F10], define $G^{-1\mathrm{/}2}$ spectrally on $\mathrm{supp}G$ and as zero on $\ker G$, and set ${\mathrm{\Lambda }}_{m,l}=G^{-1\mathrm{/}2}{\mathrm{\Gamma }}_{m,l}{G}^{-1\mathrm{/}2}$ and ${\mathrm{\Lambda }}_{\mathrm{\perp }}=I-{\mathrm{\Pi }}_{\mathrm{supp}G}$. These are nonnegative and ${\sum }_{m,l}{\mathrm{\Lambda }}_{m,l}+{\mathrm{\Lambda }}_{\mathrm{\perp }}=G^{-1\mathrm{/}2}G{G}^{-1\mathrm{/}2}+I-{\mathrm{\Pi }}_{\mathrm{supp}G}=I\mathrm{.}$ Assign ${\mathrm{\Lambda }}_{\mathrm{\perp }}$ to the pair $(1,1)$; the resulting pair effects still sum to the identity, and coarsening them by $m$ gives Bob's message POVM. For each pair, the Hayashi–Nagaoka inequality [F4] with $S={\mathrm{\Gamma }}_{m,l}$ and $T={\sum }_{(m^{\mathrm{\prime }},l^{\mathrm{\prime }})\mathrm{\ne }(m,l)}{\mathrm{\Gamma }}_{m^{\mathrm{\prime }},l^{\mathrm{\prime }}}$ bounds its pair-decoding error by twice its own projector failure plus four times its interference sum. The assigned failure effect only decreases the error for its designated pair. The estimate is valid without a uniform per-word projector-acceptance hypothesis: only its average is needed here. Specifically, $\mathbb{E}[1-\mathrm{tr}({\mathrm{\Gamma }}_{m,l}{\beta }_{X^n(m,l)})]\le {\epsilon }_n+2\sqrt{{\epsilon }_n},$ where ${\epsilon }_n\le C{e}^{-cn}$ bounds both averaged projector failures. This follows by $\mathrm{tr}(Q_{x}P{\beta }_{x}P)\ge \mathrm{tr}(Q_x{\beta }_x)-\mathrm{\parallel }P{\beta }_{x}P-{\beta }_x{\mathrm{\parallel }}_1$ and the gentle-measurement bound followed by Jensen. For two different independent codewords, the spectral eigenvalue and rank bounds give $\mathbb{E}\mathrm{tr}({\mathrm{\Gamma }}_j{\beta }_{X_i^n})=\mathrm{tr}[(P\mathbb{E}{Q}_{X_j^n}P)(P{\bar{\beta }}^{\otimes n}P)]\le e^{-n(S(\bar{\beta })-\eta )}{e}^{n({\sum }_x{p}_{x}S({\beta }_x)+\eta )}=e^{-n({\chi }_B-2\eta )}\mathrm{.}$ The equality uses the support of the first factor; the inequality uses positivity of the operator-order difference, not commutation of the two factors. Thus expected pair-error is at most $2({\epsilon }_n+2\sqrt{{\epsilon }_n})+4M_n{L}_n{e}^{-n({\chi }_B-2\eta )},$ which decays exponentially. Coarsening the pair POVM to its message label cannot increase message error.

For Eve let $P,Q_{x^n}$ denote the spectral projectors for her ensemble, put $s_E={\sum }_x{p}_{x}S({ϵ}_x)$, and define ${\xi }_{x^n}=P{Q}_{x^n}{ϵ}_{x^n}{Q}_{x^n}P,\theta =\mathbb{E}{\xi }_{X^n}\mathrm{.}$ These operators satisfy $0⪯{\xi }_{x^n}⪯e^{-n(s_E-\eta )}P$, $\mathrm{tr}{\xi }_{x^n}\le 1$, and $\mathrm{rank}P\le e^{n(S(\overline{ϵ})+\eta )}$. Their average truncation error obeys $\mathbb{E}\mathrm{\parallel }{ϵ}_{X^n}-{\xi }_{X^n}{\mathrm{\parallel }}_1\le {\zeta }_n:={\epsilon }_n+2\sqrt{{\epsilon }_n}\mathrm{.}$ Indeed the conditional projector commutes with its state, so its trace-norm loss equals its failure probability. Its subnormalized output $A$ satisfies $0⪯A⪯{ϵ}_{x^n}$, and the global-projection loss is at most $2\sqrt{\mathrm{tr}[(I-P)A]}\le 2\sqrt{\mathrm{tr}[(I-P){ϵ}_{x^n}]}$. The subnormalized gentle estimate follows by applying the normalized lemma to $A\mathrm{/}\mathrm{tr}A$ and multiplying back (the zero operator is immediate). Jensen gives the stated averaged bound.

For the $L_n$ independent words of a fixed message, centered Hilbert–Schmidt variance gives $\mathbb{E}\mathrm{tr}\left[{(\frac{1}{L_n}{\sum }_l{\xi }_{X^n(m,l)}-\theta )}^2\right]=\frac{1}{L_n}\{\mathbb{E}\mathrm{tr}({\xi }_{X^n}^2)-\mathrm{tr}({\theta }^2)\}\le \frac{e^{-n(s_E-\eta )}}{L_n}\mathrm{.}$ All cross terms have zero expectation by independence. The difference is supported in $P$. Cauchy–Schwarz on singular values and Jensen imply $\mathbb{E}{\parallel \frac{1}{L_n}{\sum }_l{\xi }_{X^n(m,l)}-\theta \parallel }_1\le \sqrt{\frac{e^{n({\chi }_E+2\eta )}}{L_n}}\mathrm{.}$ Together with $\mathrm{\parallel }\theta -{\overline{ϵ}}^{\otimes n}{\mathrm{\parallel }}_1\le {\zeta }_n$, this proves $\mathbb{E}{\parallel {\tau }_{m,n}-{\overline{ϵ}}^{\otimes n}\parallel }_1\le 2{\zeta }_n+\sqrt{\frac{e^{n({\chi }_E+2\eta )}}{L_n}},{\tau }_{m,n}=({\mathcal{N}}^c{)}^{\otimes n}({\rho }_{m,n})\mathrm{.}$ The right side decays exponentially and is independent of $m$. Consequently the expectation of the sum of average decoding error and average half-trace distance from this target decays exponentially. The codebook sample space is finite, so order the finite word/codebook alphabet and select its first deterministic codebook whose sum is at most twice its expectation (or zero if that expectation vanishes). This defines a sequence without countable choices. For that codebook let $a_n=M_n^{-1}{\sum }_{m}T({\tau }_{m,n},{\overline{ϵ}}^{\otimes n})$. The proved quantitative entropy estimate yields ${\mathrm{\ell }}_n\le 2a_{n}n\ln \dim E+2g(a_n)⟶0.$ This establishes the required unnormalized Holevo strong secrecy; a mere limit of trace distance without this dimension/error-rate estimate would not suffice. The average decoding error also tends to zero, while $n^{-1}\ln M_n\to R$. If the ensemble's private information is nonpositive it gives no positive lower bound; rate zero remains achievable with one message.