Position-based coding and convex splitting for private communication over quantum channels

摘要

The classical-input quantum-output (cq) wiretap channel is a communicationmodel involving a classical sender $X$, a legitimate quantum receiver $B$, anda quantum eavesdropper $E$. The goal of a private communication protocol thatuses such a channel is for the sender $X$ to transmit a message in such a waythat the legitimate receiver $B$ can decode it reliably, while the eavesdropper$E$ learns essentially nothing about which message was transmitted. The$\varepsilon $-one-shot private capacity of a cq wiretap channel is equal tothe maximum number of bits that can be transmitted over the channel, such thatthe privacy error is no larger than $\varepsilon\in(0,1)$. The present paperprovides a lower bound on the $\varepsilon$-one-shot private classicalcapacity, by exploiting the recently developed techniques of Anshu,Devabathini, Jain, and Warsi, called position-based coding and convexsplitting. The lower bound is equal to a difference of the hypothesis testingmutual information between $X$ and $B$ and the "alternate" smoothmax-information between $X$ and $E$. The one-shot lower bound then leads to anon-trivial lower bound on the second-order coding rate for private classicalcommunication over a memoryless cq wiretap channel.