  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.
