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

Understand

There are different inequivalent ways to define the R\'enyi capacity of a channel for a fixed input distribution $P$.

• In a 1995 paper Csisz\'ar 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 R\'enyi capacity, defined in terms of the sandwiched quantum R\'enyi 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 \[ sc(W,R,P)=\sup_{\alpha>1}\frac{\alpha-1}{\alpha}\left[R-\chi_{\alpha}^*(W,P)\right], \] where $\chi_{\alpha}^*(W,P)$ is the $P$-weighted sandwiched R\'enyi divergence radius of the image of the channel.