Ohya, M.; Petz, D. Notes on quantum entropy. (English) Zbl 0865.46043 Stud. Sci. Math. Hung. 31, No. 4, 423-430 (1996). The von Neumann entropy of states is investigated. Let \(\omega\) be a state on the finite-dimensional \(C^*\)-algebra \(A\) with von Neumann entropy \(S(\omega)\). Let \(A_n\) and \(\omega_n\) be \(n\)-fold tensor products of \(A\) and \(\omega\), respectively. In the first part of the paper it is proved that for any \(1>\varepsilon>0\), \[ S(\omega)= \lim_{n\to\infty} {\textstyle{1\over n}} \inf\{\log Tr_nQ_n:\;Q_n\in A_n,\;\omega_n(Q_n)\leq 1-\varepsilon\}. \] It means that \(S(\omega)\) governs asymptotically the size of rather sure projections. The second part of the paper deals with von Neumann entropy for states on arbitrary \(C^*\)-algebras. Then \(S(\omega)\) is defined in a rather technical way by means of relative entropies. The authors succeeded in proving a transparent expression of the entropy \(S(\omega)\) in terms of the decomposition of \(\omega\) into \(\sigma\)-convex combinations of pure states. Reviewer: J.Hamhalter (Praha) Cited in 5 Documents MSC: 46L30 States of selfadjoint operator algebras 82B10 Quantum equilibrium statistical mechanics (general) 94A17 Measures of information, entropy Keywords:convex combinations of pure states; von Neumann entropy of states; rather sure projections; relative entropies; decomposition PDFBibTeX XMLCite \textit{M. Ohya} and \textit{D. Petz}, Stud. Sci. Math. Hung. 31, No. 4, 423--430 (1996; Zbl 0865.46043)