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 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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Donald, M.J. On the relative entropy. Commun.Math. Phys. 105, 13–34 (1986). https://doi.org/10.1007/BF01212339
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DOI: https://doi.org/10.1007/BF01212339