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On Composite Quantum Hypothesis Testing

Berta, Mario and Brandão, Fernando G. S. L. and Hirche, Christoph (2021) On Composite Quantum Hypothesis Testing. Communications in Mathematical Physics, 385 (1). pp. 55-77. ISSN 0010-3616. doi:10.1007/s00220-021-04133-8.

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We extend quantum Stein’s lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ρ^(⊗n) against convex combinations of quantum states σ^(⊗n) can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy—featuring an explicit and universal recovery map.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Berta, Mario0000-0002-0428-3429
Brandão, Fernando G. S. L.0000-0003-3866-9378
Hirche, Christoph0000-0001-9265-827X
Additional Information:© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Received 03 April 2020; Accepted 31 May 2021; Published 10 June 2021. We thank an anonymous referee for extensive feedback and pointing out detailed solutions to multiple errors in previous versions of this manuscript. This work was completed prior to MB and FB joining the AWS Center for Quantum Computing. CH acknowledges support from the VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059), the Spanish MINECO, project FIS2013-40627-P, FIS2016-80681-P (AEI/FEDER, UE) and FPI Grant No. BES-2014-068888, as well as by the Generalitat de Catalunya, CIRIT project no. 2014-SGR-966.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Villum Foundation10059
Ministerio de Economía, Industria y Competitividad (MINECO)FIS2013-40627-P
Ministerio de Economía, Industria y Competitividad (MINECO)FIS2016-80681-P
Ministerio de Economía, Industria y Competitividad (MINECO)BES-2014-068888
Generalitat de Catalunya2014-SGR-966
Issue or Number:1
Record Number:CaltechAUTHORS:20190801-134523759
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Official Citation:Berta, M., Brandão, F.G.S.L. & Hirche, C. On Composite Quantum Hypothesis Testing. Commun. Math. Phys. 385, 55–77 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:97596
Deposited By: George Porter
Deposited On:01 Aug 2019 21:57
Last Modified:28 Jun 2021 18:31

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