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arXiv:1504.06595 · SIAM Journal on Optimization

Positive Maps and Separable Matrices

Jiawang Nie, Xinzhen Zhang

Published Jan 1, 2016• 23 citations
arXivHTMLPDF

Abstract

A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite matrices. This paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not. We propose numerical algorithms, based on Lasserre's type of semidefinite relaxations, for solving such questions. To check the positivity of a linear map, we construct a hierarchy of semidefinite relaxations for minimizing the associated biquadratic form over the unit spheres. We show that the positivity can be detected by solving a finite number of such semidefinite relaxations. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability.

DOI
10.1137/15m1018514

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