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On the relative entropy
Authors:Matthew J. Donald
Affiliation:(1) 18 Holland Street, CB4 3DL Cambridge, Great Britain
Abstract:ForA any subset of bernou(hamilt) (the bounded operators on a Hilbert space) containing the unit, and sgr and rgr restrictions of states on bernou(hamilt) toA, entA(sgr|rgr)—the entropy of sgr relative to rgr given the information inA—is defined and given an axiomatic characterisation. It is compared with entAA(sgr|rgr)—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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