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

Fumio Hiai, D. Petz

Research output: Chapter in Book/Report/Conference proceedingChapter

1 Citation (Scopus)

Abstract

Umegaki's relative entropy S(ω, φ) = Tr Dω(log Dω − log Dφ) (of states ω and φ with density operators Dω and Dφ, 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 Tr A log AB ≥ Tr A(log A + log B) is obtained for positive definite matrices A and B.

Original languageEnglish
Title of host publicationAsymptotic Theory of Quantum Statistical Inference: Selected Papers
PublisherWorld Scientific Publishing Co.
Pages43-63
Number of pages21
ISBN (Print)9789812563071, 9812560157, 9789812560155
DOIs
Publication statusPublished - Jan 1 2005

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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  • Cite this

    Hiai, F., & Petz, D. (2005). The proper formula for relative entropy and its asymptotics in quantum probability. In Asymptotic Theory of Quantum Statistical Inference: Selected Papers (pp. 43-63). World Scientific Publishing Co.. https://doi.org/10.1142/9789812563071_0004