### 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, ent_{A}(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 ent^{S}_{A}(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 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.

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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