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Maps on states preserving generalized entropy of convex combinations. (English) Zbl 1370.15028
Summary: Let $$S(H)$$ be the set of all linear positive-semidefinite self-adjoint Trace-one operators (states) on $$H$$ where $$H$$ is an at least two-dimensional finite-dimensional real or complex Hilbert space or at least three-dimensional left quaternionic Hilbert space of dimension $$n$$. Given a strictly convex function $$f : [0, 1] \mapsto \mathbb{R}$$, for any $$\rho \in S(H)$$ we define $$F(\rho) = \sum_i f(\lambda_i)$$, where $$\lambda_1, \lambda_2, \ldots, \lambda_n$$ are the eigenvalues of $$\rho$$ counted with multiplicities. In this note, we completely describe maps $$\phi : S(H) \rightarrow S(H)$$ having the property $$F(t \rho +(1 - t) \sigma) = F(t \phi(\rho) +(1 - t) \phi(\sigma))$$ for all $$t \in [0, 1]$$ and every $$\rho, \sigma \in S(H)$$. It turns out that $$\phi(\rho) = U \rho U^\ast$$, $$\rho \in S(H)$$, where $$U$$ is a real-linear isometry of $$H$$. Note that there is no surjectivity assumption and that our result in particular improves the description of maps preserving the von Neumann entropy of convex combinations of states in the complex Hilbert space. It can as well be applied to preserving Schatten or some other strictly convex norms of convex combinations of states.

##### MSC:
 15A86 Linear preserver problems 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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