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.
Highlights
In many scientific fields of interest, distinguishability is an important concept
(3) The exact one-shot distillable distinguishability of a quantum channel box is equal to the channel min-relative entropy, which is a particular case of the generalized channel divergence of Refs. [35,49]
The exact one-shot distinguishability cost of a quantum channel box is equal to the channel max-relative entropy, which is a particular case of the generalized channel divergence of Refs. [35,49] and explored in more detail in Refs. [16,48]
Summary
In many scientific fields of interest, distinguishability is an important concept. More generally, it can be considered as a resource in that it allows for making decisions, and the more distinguishable that two possibilities are, the easier and faster it is to make a decision.In a recent paper, we formalized the notion of distinguishability as a resource by developing the resource theory of asymmetric distinguishability in detail [1], following the original proposal from Refs. [2,3]. In many scientific fields of interest, distinguishability is an important concept. It can be considered as a resource in that it allows for making decisions, and the more distinguishable that two possibilities are, the easier and faster it is to make a decision. [2,3] This resource theory demonstrates that distinguishability is truly a fundamental resource that can be manipulated and interconverted into different forms. The benefit of developing this resource theory is that, can fundamental tasks such as quantum hypothesis testing [4,5,6,7,8,9] be recast into an intuitive approach based on resource-theoretic thinking, and new information process-
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