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Notes on quantum entropy. (English) Zbl 0865.46043

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.

MSC:

46L30 States of selfadjoint operator algebras
82B10 Quantum equilibrium statistical mechanics (general)
94A17 Measures of information, entropy
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