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On Composite Quantum Hypothesis Testing

Berta, Mario and Brandão, Fernando G. S. L. and Hirche, Christoph (2017) On Composite Quantum Hypothesis Testing. . (Unpublished)

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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) is given by a regularized quantum relative entropy distance 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 a novel 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 only using elementary properties of quantum entropy. Finally, our findings also imply an improved Markov type lower bound on the quantum conditional mutual information in terms of the regularized quantum relative entropy - featuring an explicit and universal recovery map.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Berta, Mario0000-0002-0428-3429
Brandão, Fernando G. S. L.0000-0003-3866-9378
Additional Information:MB acknowledges funding by the SNSF through a fellowship. CH acknowledges support by 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
Swiss National Science Foundation (SNSF)UNSPECIFIED
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
Record Number:CaltechAUTHORS:20190801-134523759
Persistent URL:
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:04 Jun 2020 10:14

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