  This paper develops the resource theory of asymmetric distinguishability for
quantum channels, generalizing the related resource theory for states
[arXiv:1010.1030; arXiv:1905.11629]. The key constituents of the channel
resource theory are quantum channel boxes, consisting of a pair of quantum
channels, which can be manipulated for free by means of an arbitrary quantum
superchannel (the most general physical transformation of a quantum channel).
One main question of the resource theory is the approximate channel box
transformation problem, in which the goal is to transform an initial channel
box (or boxes) to a final channel box (or boxes), while allowing for an
asymmetric error in the transformation. The channel resource theory is richer
than its counterpart for states because there is a wider variety of ways in
which this question can be framed, either in the one-shot or $n$-shot regimes,
with the latter having parallel and sequential variants. As in our prior work
[arXiv:1905.11629], we consider two special cases of the general channel box
transformation problem, known as distinguishability distillation and dilution.
For the one-shot case, we find that the optimal values of the various tasks are
equal to the non-smooth or smooth channel min- or max-relative entropies, thus
endowing all of these quantities with operational interpretations. In the
asymptotic sequential setting, we prove that the exact distinguishability cost
is equal to the channel max-relative entropy and the distillable
distinguishability is equal to the amortized channel relative entropy of
[arXiv:1808.01498]. This latter result can also be understood as a solution to
Stein's lemma for quantum channels in the sequential setting. Finally, the
theory simplifies significantly for environment-seizable and classical--quantum
channel boxes.
