= One-shot classical-quantum coding converse
{title2=$(1-\epsilon)\log_2k\leq I(X:Q)+1$}
For $k$ equiprobable quantum code states with average decoding error $\epsilon<1$, <data-processing inequality for quantum relative entropy> applied to the <binary test for quantum decoding success> gives
$$
I(X:Q)\geq(1-\epsilon)\log_2k-h_2(\epsilon)-\epsilon\log_2(1-1/k)\geq(1-\epsilon)\log_2k-1
$$
for $k\geq2$. Here $h_2$ is <binary entropy>. The case $k=1$ is trivial. Taking the supremum over allowed input distributions yields a code-size bound in terms of the largest <Holevo quantity>; at $\epsilon=1$ the undivided inequality is vacuous.
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