Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions

Abstract

There are different inequivalent ways to define the Rényi capacity of a channel for a fixed input distribution $P$. In a 1995 paper 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 classical-quantum channel coding. Denoting the constant composition strong converse exponent for a memoryless classical-quantum channel $W$ with composition $P$ and rate $R$ as $sc(W,R,P)$, our main result is that \[ sc(W,R,P)=\sup_{α>1}\frac{α-1}α\left[R-χ_α^*(W,P)\right], \] where ${\chi }_{\alpha }^{\ast }(W,P)$ is the $P$-weighted sandwiched Rényi divergence radius of the image of the channel.

46 pages. V7: Added the strong converse exponent with cost constraint

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• The Quantum Chernoff Bound
• Coding Theorem and Strong Converse for Quantum Channels
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• Matrix versions of the Hellinger distance
• From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture (New title)
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