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


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.


Entropy Neural Network Statistical Physic Complex System Nonlinear Dynamics 
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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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