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Communications in Mathematical Physics

, Volume 143, Issue 1, pp 99–114 | Cite as

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

  • Fumio Hiai
  • Dénes Petz
Article

Abstract

Umegaki's relative entropyS(ω,ϕ)=TrDω(logDω−logDϕ) (of states ω and ϕ with density operatorsDω andDϕ, 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.

Keywords

Entropy Neural Network Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Fumio Hiai
    • 1
  • Dénes Petz
    • 2
  1. 1.Department of MathematicsIbaraki UniversityMito, IbarakiJapan
  2. 2.Mathematical InstituteHungarian Academy of SciencesBudapestHungary

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