On the relative entropy

ForA any subset of () (the bounded operators on a Hilbert space) containing the unit, and and restrictions of states on () toA, ent_A(|)—the entropy of relative to given the information inA—is defined and given an axiomatic characterisation. It is compared with ent_A^A(|)—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.