Mathematical Inequalities & Applications

Linear maps of positive partial transpose matrices and singular value inequalities

Xiaohui Fu, Pan-Shun Lau, Tin-Yau Tam

Published Jan 1, 2020• 2 citations

Abstract

Linear maps : M n M k are called m -PPT if [(A i j )] m i, j=1 are positive partial transpose matrices for all positive semi-definite matrices

DOI: 10.7153/mia-2020-23-104

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Mathematical Inequalities & Applications

Linear maps of positive partial transpose matrices and singular value inequalities

Xiaohui Fu, Pan-Shun Lau, Tin-Yau Tam

Published Jan 1, 2020• 2 citations

Publisher

Abstract

Linear maps : M n M k are called m -PPT if [(A i j )] m i, j=1 are positive partial transpose matrices for all positive semi-definite matrices

DOI: 10.7153/mia-2020-23-104

Related papers

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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Kennett L. Dela Rosa, Iana Angela C. Fajardo, Cedie B. Reyes

Given $α\in \mathbb R$ and $ B_1,\ldots,B_q\in M_{m,n}$, consider the map \[Φ^{\pm}_{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on $α$ and $B_1,\ldots,B_q$ are given that guarantee $Φ_{α,\mathfrak{B}}^\pm$ is $k$ -positive, $k$ -copositive, and $k$ -PPT, respectively. As application, known results about the map $α\mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.Given $α\in \mathbb R$ and $ B_1,\ldots,B_q\in M_{m,n}$, consider the map \[Φ^{\pm}_{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on $α$ and $B_1,\ldots,B_q$ are given that guarantee $Φ_{α,\mathfrak{B}}^\pm$ is $k$-positive, $k$-copositive, and $k$-PPT, respectively. As application, known results about the map $α\mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.

arXiv:2609.07260Sep 7, 20260 citations

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.

Given

α\in \mathbb R

and $ B_1,\ldots,B_q\in M_{m,n}$, consider the map \[Φ^{\pm}_{α,\mathfrak{B}}(X)=α\mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on

α

and

B_1,\ldots,B_q

are given that guarantee

Φ_{α,\mathfrak{B}}^\pm

k

-positive,

k

-copositive, and

k

-PPT, respectively. As application, known results about the map $α\mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.