On the Relative Entropy.

Communications in Mathematical Physics, Volume 105, pages 13-34 (1986)

Abstract For A 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) to A, entA(σ|ρ) — the entropy of σ relative to ρ given the information in A — is defined and given an axiomatic characterisation. It is compared with entSA(σ|ρ) — the relative entropy introduced by Umegaki and generalised by various authors — which is defined only for A an algebra. It is proved that ent and entS 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 entS: monotonicity, concavity, w* upper semicontinuity, etc.

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Matthew J. Donald

The Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, Great Britain.

e-mail : mjd1014@cam.ac.uk

home page: http://people.bss.phy.cam.ac.uk/~mjd1014