Characterization of the relative entropy of states of matrix algebras (Q1206297)

Umegaki introduced the relative entropy of two density matrices as \(S(D_ 1,D_ 2)=\text{Tr }D_ 1(\log D_ 1-\log D_ 2)\). In the paper this information quantity is characterized by the following properties: (1) Conditional expectation property; (2) Direct sum property; (3) Invariance under automorphisms, (4) Measurability. These postulates determine relative entropy up to a multiplicative factor. Extension to the setting of nuclear \(C^*\)-algebras is found in the recent monograph: \textit{M. Ohya} and \textit{D. Petz}, ``Quantum entropy and its use'' (1993).