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Isometries and relative entropy preserving maps on density operators. (English) Zbl 1241.47034
Let \(H\) be a finite-dimensional complex Hilbert space. A density operator on \(H\) is a positive operator whose trace is \(1\). Let \(S(H)\) denote the set of all density operators on \(H\) and let \(M(H)\) be the subset of all invertible density operators. The first result of the paper asserts that, if \(\phi: S(H) \to S(H)\) is an isometry with respect to the metric which comes from the von Neumann-Schatten \(p\)-norm, then there exists either a unitary or an antiunitary operator \(U\) such that \(\phi(A)=UAU^\ast\), for all \(A\in S(H)\). There are two more results related to preservers in the paper. The transformations on \(S(H)\) and \(M(H)\) are studied which preserve different relative entropies. These results are continuation of an earlier work of the first author.

MSC:
47B49 Transformers, preservers (linear operators on spaces of linear operators)
47N50 Applications of operator theory in the physical sciences
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