There are different inequivalent ways to define the Renyi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csiszar 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 Renyi capacity, defined in terms of the sandwiched quantum Renyi 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 Renyi divergence radius of the image of the channel.
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