Resource theory of asymmetric distinguishability for quantum channels

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Resource theory of asymmetric distinguishability for quantum channels

Xin Wang and Mark M. Wilde

Phys. Rev. Research 1, 033169 – Published 11 December 2019

Abstract

This paper develops the resource theory of asymmetric distinguishability for quantum channels, generalizing the related resource theory for states [Matsumoto, arXiv:1010.1030, Wang and Wilde, Phys. Rev. Research 1, 033170 (2019)]. 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 [Wang and Wilde, Phys. Rev. Research 1, 033170 (2019)], 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 nonsmooth 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 [Berta, Hirche, Kaur, and Wilde, 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.

Received 20 July 2019

DOI:https://doi.org/10.1103/PhysRevResearch.1.033169

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum channels Quantum information processing Resource theories

Quantum Information, Science & Technology

Authors & Affiliations

Xin Wang^1,2,* and Mark M. Wilde ^3,†

¹Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
²Institute for Quantum Computing, Baidu Research, Beijing 100193, China
³Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA

^*wangxinfelix@gmail.com
^†mwilde@lsu.edu

Resource theory of asymmetric distinguishability

Xin Wang and Mark M. Wilde

Phys. Rev. Research 1, 033170 (2019)

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Vol. 1, Iss. 3 — December - December 2019

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Images

Figure 1
The figure depicts the transformation of a channel $N_{A \to B}$ by a superchannel $Θ_{(A \to B) \to (C \to D)}$ , the latter of which consists of the preprocessing channel $E_{C \to A M}$ and the postprocessing channel $D_{B M \to D}$ .
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Figure 2
A depiction of the state $ρ_{R_{n} B_{n}}$ in Eq. (107) with $n = 3$ , which results from the interaction of a three-round quantum strategy $N^{(3)}$ with a co-strategy.
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Figure 3
A depiction of the transformation of a three-round quantum strategy $N^{(3)}$ to a three-round quantum strategy $K^{(3)}$ by a physical transformation $Θ^{(3 \to 3)}$ consisting of the channels $F^{1}, ..., F^{6}$ , along with the pairing of the transformed strategy with a quantum co-strategy.
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Figure 4
Depiction of the physical transformation $Θ^{(n \to m)}$ that converts the three-round strategy $N^{(3)}$ to the three-round strategy $K^{(3)}$ . The physical transformation $Θ^{(n \to m)}$ consists of the channels $F^{1}, ..., F^{6}$ . A discriminator could in principle then perform a co-strategy to distinguish the simulation in the top part of the figure from the ideal implementation of the strategy $K^{(3)}$ in the bottom part of the figure. We demand that the absolute deviation in probability between any measurement outcome in the top part be no larger than $ɛ$ when compared to the same from the bottom part, i.e., that the normalized strategy distance be no larger than $ɛ$ .
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Figure 5
Depiction of the physical transformation $Θ^{(n \to m)}$ that converts the three-round strategy $M^{(3)}$ to the three-round strategy $L^{(3)}$ . The physical transformation $Θ^{(n \to m)}$ is the same as that given in Fig. 4 and consists of the channels $F^{1}, ..., F^{6}$ . A discriminator could in principle then perform a strategy to distinguish the simulation in the top part of the figure from the ideal implementation of the three-round strategy $L^{(3)}$ in the bottom part of the figure. We demand that the absolute deviation in probability between any measurement outcome in the top part be exactly equal to zero when compared to the same from the bottom part, i.e., that the normalized strategy distance be equal to zero, so that the simulation is perfect in this case.
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Figure 6
Depiction of a sequential channel box transformation protocol. Three sequential uses of the channel $N$ are converted approximately to three sequential uses of the channel $K$ , while three sequential uses of the channel $M$ are converted exactly to three sequential uses of the channel $L$ . This is a special case of a strategy box transformation protocol, as depicted in Figs. 4 and 5.
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