Divergence Radii and the Strong Converse Exponent of Classical-Quantum Channel Coding With Constant Compositions

There are different inequivalent ways to define the Rényi capacity of a channel for a fixed input distribution. In [IEEE Transactions on Information Theory, 41(1):26-34, 1995], Csiszár has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényi divergences, has the same operational interpretation in the strong converse problem of constant composition classical-quantum channel coding.

I. Introduction

A classical-quantum channel models a device with a set of possible inputs , which, on an input , outputs a quantum system with finite-dimensional Hilbert space in state . The channel is called classical if all output states commute, and hence can be represented as probability distributions on some set ; the interpretation of this is that the channel outputs the symbol with probability . We will only deal with i.i.d. (independent and identically distributed) channels, meaning that when the channel is used sequentially, it emits the state on an input sequence .