Towards the ultimate limits of quantum channel discrimination

Abstract: This note studies the difficulty of discriminating quantum channels under operational regimes. First, we make a conjecture on the exponentially strong converse of quantum channel hypothesis testing under coherent strategies, meaning that any strategy to make the Type II error decays with an exponent larger than the regularized channel relative entropy will unavoidably result in the Type I error converging to one exponentially fast in the asymptotic limit. This conjecture will imply the desirable quantum channel Stein's Lemma and the continuity of the regularized (amortized) Sandwiched R\'{e}nyi channel divergence at $\alpha=1$. We also remark that there was a gap in the proof of the above conjecture in our previous arXiv version. Such gap exists since a lemma basically comes from [Brandao and Plenio, 2010] was found to be false. Second, we develop a framework to show the interplay between the strategies of channel discrimination, the operational regimes, and variants of channel divergences. This framework systematically underlies the operational meaning of quantum channel divergences in quantum channel discrimination. Our work makes an attempt towards understanding the ultimate limit of quantum channel discrimination, as well as its connection to quantum channel divergences in the asymptotic regime.