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The proper formula for relative entropy and its asymptotics in quantum probability

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Abstract

Umegaki's relative entropyS(ω,ϕ)=TrD ω(logD ω−logD ϕ) (of states ω and ϕ with density operatorsD ω andD ϕ, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality TrA logAB ≧TrA(logA+logB) is obtained for positive definite matricesA andB.

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Authors and Affiliations

  1. Department of Mathematics, Ibaraki University, 310, Mito, Ibaraki, Japan

    Fumio Hiai

  2. Mathematical Institute, Hungarian Academy of Sciences, P.O. Box 127, H-1364, Budapest, Hungary

    Dénes Petz

Authors
  1. Fumio Hiai
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  2. Dénes Petz
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Communicated by H. Araki

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Hiai, F., Petz, D. The proper formula for relative entropy and its asymptotics in quantum probability. Commun.Math. Phys. 143, 99–114 (1991). https://doi.org/10.1007/BF02100287

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  • DOI: https://doi.org/10.1007/BF02100287

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