Resource theory of asymmetric distinguishability for quantum channels

Abstract

This paper develops the resource theory of asymmetric distinguishability for quantum channels, generalizing the related resource theory for states [arXiv:1006.0302, arXiv:1905]. 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 [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 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. arXiv:1907.06306v1 [quant-ph]

Chain Rule for the Quantum Relative Entropy

“…This answers an open question in the area of channel discrimination showing that, surprisingly, adaptive strategies cannot be more powerful than non-adaptive strategies [13,2,25,26]. We comment on this in Section 3.4.…”

“…Based on the measurement outcome we try to guess if G = E or G = F. The protocol is depicted in Figure 2a. It has been shown recently [25,Theorem 3] that in the asymmetric regime where we fix the type-I error to be bounded by ε, the asymptotic optimal rate of the type-II error exponent is given by D reg (E F), when ε goes to 0.…”

“…More precisely, for an arbitrary state ρ AR and based on the outcome try to guess if G = E or G = F. The strategy is depicted in Figure 2b. The asymptotically optimal rate of the type-II error exponent for this strategy is given by D A (E F) [25,Theorem 6].…”

Resource theory of asymmetric distinguishability

“…Going forward from here, there are many interesting directions to pursue. The resource theory of asymmetric distinguishability for quantum channels will be developed in a future work [WW19]. The main constituents consist of a channel box (N , M), for quantum channels N and M, as the basic objects of manipulation, superchannels [CDP08] as the free operations, and bits of asymmetric distinguishability as the fundamental currency.…”

Resource Preservability

“…Notably, resource theories for quantum channels (i.e. completelypositive trace-preserving maps [12]) and related topics have drawn much attention recently [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59].…”