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arXiv:1409.3673 · International Journal of Mathematics

Around Shannon's interpretation for entropy-preserving stochastic averages

Marie Choda

Published Sep 1, 2014• 8 citations
arXivHTMLPDF

Abstract

Let Φ be a positive unital Tr-preserving map on Mn(ℂ). We give various characterizations for Φ to preserve the von Neumann entropy of a state ρ on Mn(ℂ). Among others, it is given by that Φ behaves to ρ as a *-automorphism. This is also equivalent to that the entropy of the stochastic matrix arising from {ρ, Φ} is zero.
DOI
10.1142/s0129167x14500967

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Von Neumann defined the informational entropy of a density matrix ρ by the expression S(ρ)=−tr(ρ ln ρ). [J. Von Neumann, Die Mathematischen Grundlagen der Quantummechanik (Springer-Verlag, Berlin, 1932)]. Here, starting from the definitions of Shannon entropy and of Renyi entropy of random variables, and using some (reasonable) rules of inference, two expressions are obtained for the quantum entropy of a given square (deterministic) matrix, irrespective of any probabilistic framework and/or any randomization technique. These definitions apply directly to linear operators on Hilbert spaces even when they are not of positive trace class, and they contain Von Neumann entropy as a special case. These new measures of uncertainty could provide new approaches to some problems related to structure complexity.
Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten ppp -norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a C∗C^{\ast }C∗ -algebra.
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