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arXiv:1006.4155

Continuity condition for concave functions on convex $μ$-compact sets and its applications in quantum physics

Максим Евгеньевич Широков

Published Jun 21, 2010• 1 citation
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Abstract

A method of proving local continuity of concave functions on convex set possessing the $μ$-compactness property is presented. This method is based on a special approximation of these functions. The class of $μ$-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which particular results well known for compact sets can be generalized. Applications of the obtained continuity conditions to analysis of different entropic characteristics of quantum systems and channels are considered.

DOI
10.48550/arxiv.1006.4155

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Continuity condition for concave functions on convex μμμ-compact sets and its applications in quantum physics
A method of proving local continuity of concave functions on convex set possessing the μμμ-compactness property is presented. This method is based on a special approximation of these functions. The class of μμμ-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which particular results well known for compact sets can be generalized. Applications of the obtained continuity conditions to analysis of different entropic characteristics of quantum systems and channels are considered.
The universal method of proving continuity of the von Neumann entropy on subsets of positive trace-class operators are considered. It makes possible to re-derive the known conditions of continuity of the entropy in the more general forms and to obtain the several new conditions. This method is based on the special approximation of the von Neumann entropy by the increasing sequence of concave continuous unitary invariant functions. Existence of this approximation is a corollary of the general property of the set of quantum states as a convex topological space called the strong stability property and considered in the first part of the paper.
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