On Composite Quantum Hypothesis Testing

Department of Mathematical Sciences

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On Composite Quantum Hypothesis Testing. / Berta, Mario; Brandão, Fernando G.S.L.; Hirche, Christoph.

In: Communications in Mathematical Physics, Vol. 385, No. 1, 2021, p. 55-77.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Berta, M, Brandão, FGSL & Hirche, C 2021, 'On Composite Quantum Hypothesis Testing', Communications in Mathematical Physics, vol. 385, no. 1, pp. 55-77. https://doi.org/10.1007/s00220-021-04133-8

APA

Berta, M., Brandão, F. G. S. L., & Hirche, C. (2021). On Composite Quantum Hypothesis Testing. Communications in Mathematical Physics, 385(1), 55-77. https://doi.org/10.1007/s00220-021-04133-8

Vancouver

Berta M, Brandão FGSL, Hirche C. On Composite Quantum Hypothesis Testing. Communications in Mathematical Physics. 2021;385(1):55-77. https://doi.org/10.1007/s00220-021-04133-8

Author

Berta, Mario ; Brandão, Fernando G.S.L. ; Hirche, Christoph. / On Composite Quantum Hypothesis Testing. In: Communications in Mathematical Physics. 2021 ; Vol. 385, No. 1. pp. 55-77.

Bibtex

@article{366f53c52636402ca929f760ed5649a0,

title = "On Composite Quantum Hypothesis Testing",

abstract = "We extend quantum Stein{\textquoteright}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{\textquoteright}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.",

author = "Mario Berta and Brand{\~a}o, {Fernando G.S.L.} and Christoph Hirche",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s).",

year = "2021",

doi = "10.1007/s00220-021-04133-8",

language = "English",

volume = "385",

pages = "55--77",

journal = "Communications in Mathematical Physics",

issn = "0010-3616",

publisher = "Springer",

number = "1",

}

RIS

TY - JOUR

T1 - On Composite Quantum Hypothesis Testing

AU - Berta, Mario

AU - Brandão, Fernando G.S.L.

AU - Hirche, Christoph

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85107461338&partnerID=8YFLogxK

U2 - 10.1007/s00220-021-04133-8

DO - 10.1007/s00220-021-04133-8

M3 - Journal article

AN - SCOPUS:85107461338

VL - 385

SP - 55

EP - 77

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -

ID: 276379985