Properties of the relative entropy of states of von Neumann algebras
type: Item
instance of: scholarly article
title: Properties of the relative entropy of states of von Neumann algebras
zbMATH Open document ID: 0656.46049
DOI: 10.1007/BF01949125
published in: Acta Mathematica Hungarica
publication date: 1986-01-01T00:00:00Z
review text: 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 \(F\colon a'\xi\mapsto a'{}^*\eta.\}\)
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L30
zbMATH DE Number: 4071683
zbMATH Keywords: relative entropy
zbMATH Keywords: noncommutative extension of Kullback-Leibler information
zbMATH Keywords: relative entropy of states on von Neumann algebras
zbMATH Keywords: superadditivity of relative entropy
author: Dénes Petz
reviewed by: Dénes Petz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4071683