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

Fumio Hiai, D. Petz

Research output: Contribution to journalArticle

224 Citations (Scopus)

Abstract

Umegaki's relative entropy S(ω,φ{symbol})=Tr Dω(log Dω-log Dφ{symbol}) (of states ω and φ{symbol} with density operators Dω and Dφ{symbol}, 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
Pages (from-to)99-114
Number of pages16
JournalCommunications in Mathematical Physics
Volume143
Issue number1
DOIs
Publication statusPublished - Dec 1991

Fingerprint

Quantum Probability
Relative Entropy
entropy
Density Operator
Positive definite matrix
Hypothesis Testing
Exponent
exponents
operators
matrices

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

The proper formula for relative entropy and its asymptotics in quantum probability. / Hiai, Fumio; Petz, D.

In: Communications in Mathematical Physics, Vol. 143, No. 1, 12.1991, p. 99-114.

Research output: Contribution to journalArticle

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