# Resource theory of asymmetric distinguishability for quantum channels

## TL;DR

This paper develops a resource theory for the asymmetric distinguishability of quantum channels, providing operational interpretations for related entropic quantities and analyzing transformation tasks in various regimes.

## Contribution

It generalizes the resource theory of state distinguishability to channels, introduces operational meanings for channel min- and max-relative entropies, and analyzes asymptotic and one-shot transformation costs.

## Key findings

- Optimal one-shot distinguishability measures relate to smooth channel relative entropies.
- Asymptotic distinguishability cost equals channel max-relative entropy.
- Distillable distinguishability equals amortized channel relative entropy.

## Abstract

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.