Properties of the relative entropy of states of von Neumann algebras
The relative entropy introduced by \textit{H. Umegaki} [Kōdai Math. Sem. Rep. 14, 59--85 (1962; Zbl 0199.19706)] as the noncommutative extension of Kullback-Leibler information was generalized by \textit{H. Araki} [Publ. Res. Inst. Math. Sci. Kyoto Univ. 11, No. 3, 809--833 (1976; Zbl 0326.46031); ibid. 13, No. 1, 173--192 (1977; Zbl 0374.46055)] to the general von Neumann algebra case and by \textit{A. Uhlmann} [Commun. Math. Phys. 54, No. 1, 21--32 (1977; Zbl 0358.46026)] to the arbitrary \({}^*\)-algebra case. This paper deals with several properties of the relative entropy of states on von Neumann algebras. Although the main properties such as the monotonicity are already known from Araki's and Uhlmann's papers, the proofs here are more accessible. Also, the strong superadditivity of relative entropy is shown in a simple way. But the assumption of faithfulness of \(\phi\) in considering the relative entropy \(S(\omega,\phi)\) is somewhat restrictive. \{There are some typographical errors in the paper; for instance, \(S\colon a\xi\mapsto a^*\xi\) and \(F\colon a'\xi\mapsto a'{}^*\xi\) in Lemma 4 should be \(S\colon a\xi\mapsto a^*\eta\) and \(F\colon a'\xi\mapsto a'{}^*\eta.\}\)
noncommutative extension of Kullback-Leibler information
relative entropy of states on von Neumann algebras
scientific article; zbMATH DE number 4071683
scientific article