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Communications in Mathematical Physics

, Volume 105, Issue 1, pp 13–34 | Cite as

On the relative entropy

  • Matthew J. Donald
Article

Abstract

ForA any subset of ℬ(ℋ) (the bounded operators on a Hilbert space) containing the unit, and σ and ρ restrictions of states on ℬ(ℋ) toA, ent A (σ|ρ)—the entropy of σ relative to ρ given the information inA—is defined and given an axiomatic characterisation. It is compared with ent A A (σ|ρ)—the relative entropy introduced by Umegaki and generalised by various authors—which is defined only forA an algebra. It is proved that ent and ent S agree on pairs of normal states on an injective von Neumann algebra. It is also proved that ent always has all the most important properties known for ent S : monotonicity, concavity,w* upper semicontinuity, etc.

Keywords

Entropy Neural Network Statistical Physic Hilbert Space Normal State 
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 1986

Authors and Affiliations

  • Matthew J. Donald
    • 1
  1. 1.CambridgeGreat Britain

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