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arXiv:1012.4754

Completely positive mappings and mean matrices

Ádám Besenyei, Dėnes Petz

Published Dec 21, 2010• 0 citations
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Abstract

Some real functions f induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define linear mapping beta on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of beta in the case of a few concrete functions f. This problem has been motivated by applications in quantum information.

DOI
10.48550/arxiv.1012.4754

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Some real functions f induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define linear mapping beta on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of beta in the case of a few concrete functions f. This problem has been motivated by applications in quantum information.
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Abstract A ∗\ast∗-linear map Φ\PhiΦ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations In⊗ΦI_n\otimes \PhiIn​⊗Φ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.
Quantum monotone metric was introduced by Petz [Linear Algebra Appl. 244, 81–96 (1996)], and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always the one based on completely positive, trace preserving maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators and did not have some ideal properties such as monotonicity and convexity with respect to the positive operators. In this paper, we introduce another extension of quantum monotone metrics that have monotonicity under completely positive, trace non-increasing maps and additive noise. We prove that our extended monotone metrics can be characterized only by static operator monotone functions from few assumptions without assuming continuities of metrics. We show that our monotone metrics have some natural properties such as additivity of direct sum, convexity, and monotonicity with respect to positive operators.
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