On Composite Quantum Hypothesis Testing
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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
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TY - JOUR
T1 - On Composite Quantum Hypothesis Testing
AU - Berta, Mario
AU - Brandão, Fernando G.S.L.
AU - Hirche, Christoph
N1 - Publisher Copyright: © 2021, The Author(s).
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