Stein\u27s Lemma for Classical-Quantum Channels

It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We show that, for the discrimination of classical-quantum channels, adaptive strategies do not lead to an asymptotic advantage. As our main result, this establishes Stein\u27s lemma for classical-quantum channels. Our proofs are based on the concept of amortized distinguishability of channels, which we analyse using entropy inequalities