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arXiv:quant-ph/0610020

Parametrizations of Positive Matrices With Applications

Tseng, M. C., Hong Zhou, Viswanath Ramakrishna

Published Oct 3, 2006• 0 citations
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Abstract

This paper reviews some characterizations of positive matrices and discusses which lead to useful parametrizations. It is argued that one of them, which we dub the Schur-Constantinescu parametrization is particularly useful. Two new applications of it are given. One shows all block-Toeplitz states are PPT. The other application is to relaxation rates.

DOI
10.48550/arxiv.quant-ph/0610020

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This paper reviews some characterizations of positive matrices and discusses which lead to useful parametrizations. It is argued that one of them, which we dub the Schur-Constantinescu parametrization is particularly useful. Two new applications of it are given. One shows all block-Toeplitz states are PPT. The other application is to relaxation rates.
Abstract The purpose of this work is twofold. The first is to survey some parametrizations of positive matrices which have found applications in quantum information theory. The second is to provide some more applications of a parametrization of quantum states and channels introduced by T. Constantinescu and the last author, and thereby to provide further evidence of the utility of this parametrization. This work is dedicated to the memory of our colleague and teacher, the late Professor T. Constantinescu.
This paper mainly focuses on the class of 2×22 \times 22×2 block PPT matrices. The relationship between PPT matrices and the norm inequalities is further explored. Some properties of a non-PPT matrix in terms of its eigenvalues are investigated. Moreover, a number of useful sufficient conditions for a matrix to be PPT are provided.
A block matrix [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] is positive partial transpose (PPT) if both [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] and [AX∗XB]\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right][AX​X∗B​] are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.