Communications in Mathematical Physics

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

On the relative entropy

  • Matthew J. Donald


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.


Entropy Neural Network Statistical Physic Hilbert Space Normal State 
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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

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

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