One-shot classical-quantum coding converse ($(1-ϵ){\log }_{2}k\le I(X:Q)+1$)

For $k$ equiprobable quantum code states with average decoding error $ϵ<1$, data-processing inequality for quantum relative entropy applied to the binary test for quantum decoding success givesfor $k\ge 2$. 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 $ϵ=1$ the undivided inequality is vacuous.