AIMS Mathematics

A note on positive partial transpose blocks

Moh. Alakhrass

Published Jan 1, 2023• 3 citations

Abstract

In this article, we study the class of positive partial transpose blocks. We introduce several inequalities related to this class with an emphasis on comparing the main diagonal and off-diagonal components of a $ 2 \times 2 $ positive partial transpose block.

DOI: 10.3934/math.20231208

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A note on positive partial transpose blocks

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AIMS Mathematics

A note on positive partial transpose blocks

Moh. Alakhrass

Published Jan 1, 2023• 3 citations

Publisher

Abstract

2 \times 2

positive partial transpose block.

DOI: 10.3934/math.20231208

Related papers

A note on positive partial transpose blocks

Mohammad Alakhrass

arXiv:2112.13667Dec 22, 20210 citations

On Positive Partial Transpose Matrices

Mehmet Gumus, Jianzhen Liu, Samir Raouafi +1 more

This paper mainly focuses on the class of $2 \times 2$ 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.This paper mainly focuses on the class of $2 \times 2$ 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.

Electronic Journal of Linear AlgebraMay 13, 20201 citation

On positive and positive partial transpose matrices

İbrahim Halil Gümüş, Hamid Reza Moradi, Mohammad Sababheh

A block matrix $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ is positive partial transpose (PPT) if both $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ and $\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right]$ 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.A block matrix $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ is positive partial transpose (PPT) if both $\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]$ and $\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right]$ 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.

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2 \times 2

positive partial transpose block.

This paper mainly focuses on the class of

2 \times 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

\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]

is positive partial transpose (PPT) if both

\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right]

and

\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right]

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.