Electronic Journal of Linear Algebra

On Positive Partial Transpose Matrices

Mehmet Gumus, Jianzhen Liu, Samir Raouafi, Tin-Yau Tam

Published May 13, 2020• 1 citation

Abstract

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.

DOI: 10.13001/ela.2020.5197

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Electronic Journal of Linear Algebra

On Positive Partial Transpose Matrices

Mehmet Gumus, Jianzhen Liu, Samir Raouafi, Tin-Yau Tam

Published May 13, 2020• 1 citation

Publisher

Abstract

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.

DOI: 10.13001/ela.2020.5197

Related papers

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

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In this paper we solve the following problem. Let ${\mathcal{H}}_{nm}$ be a Hilbert space of dimension $nm$ , and let $A$ be a positive semidefinite self-adjoint linear operator on ${\mathcal{H}}_{nm}$ . Under which conditions on the spectrum has $A$ a positive partial transpose (is PPT) with respect to any partition ${\mathcal{H}}_{n}\ensuremath{\bigotimes}{\mathcal{H}}_{m}$ of the space ${\mathcal{H}}_{nm}$ as a tensor product of an $n$ -dimensional and an $m$ -dimensional Hilbert space? We show that the necessary and sufficient conditions can be expressed as a set of linear matrix inequalities (LMIs) on the eigenvalues of $A$ .In this paper we solve the following problem. Let ${\mathcal{H}}_{nm}$ be a Hilbert space of dimension $nm$, and let $A$ be a positive semidefinite self-adjoint linear operator on ${\mathcal{H}}_{nm}$. Under which conditions on the spectrum has $A$ a positive partial transpose (is PPT) with respect to any partition ${\mathcal{H}}_{n}\ensuremath{\bigotimes}{\mathcal{H}}_{m}$ of the space ${\mathcal{H}}_{nm}$ as a tensor product of an $n$-dimensional and an $m$-dimensional Hilbert space? We show that the necessary and sufficient conditions can be expressed as a set of linear matrix inequalities (LMIs) on the eigenvalues of $A$.

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Daniel Cariello

Recently it was proved that many results that are true for density matrices which are positive under partial transposition (or simply PPT), also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called symmetric with positive coefficients (or simply SPC). A natural question appeared: What is the connection between SPC matrices and PPT matrices? Is every SPC matrix PPT? Here we show that every SPC matrix is PPT in $M_2\otimes M_2$ . This theorem is a consequence of the fact that every density matrix in $M_2\otimes M_m$ , with tensor rank smaller or equal to 3, is separable. Although, in $M_3\otimes M_3$ , we present an example of SPC matrix with tensor rank 3 that is not PPT. We shall also provide a non trivial example of a family of matrices in $M_k\otimes M_k$ , in which both, the SPC and PPT properties, are equivalent. Within this family, there exists a non trivial subfamily in which the SPC property is equivalent to separability.Recently it was proved that many results that are true for density matrices which are positive under partial transposition (or simply PPT), also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called symmetric with positive coefficients (or simply SPC). A natural question appeared: What is the connection between SPC matrices and PPT matrices? Is every SPC matrix PPT? Here we show that every SPC matrix is PPT in $M_2\otimes M_2$. This theorem is a consequence of the fact that every density matrix in $M_2\otimes M_m$, with tensor rank smaller or equal to 3, is separable. Although, in $M_3\otimes M_3$, we present an example of SPC matrix with tensor rank 3 that is not PPT. We shall also provide a non trivial example of a family of matrices in $M_k\otimes M_k$, in which both, the SPC and PPT properties, are equivalent. Within this family, there exists a non trivial subfamily in which the SPC property is equivalent to separability.

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This paper mainly focuses on the class of

2 \times 2

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.

In this paper we solve the following problem. Let

{\mathcal{H}}_{nm}

be a Hilbert space of dimension

nm

, and let

A

be a positive semidefinite self-adjoint linear operator on

{\mathcal{H}}_{nm}

. Under which conditions on the spectrum has

A

a positive partial transpose (is PPT) with respect to any partition ${\mathcal{H}}_{n}\ensuremath{\bigotimes}{\mathcal{H}}_{m}$ of the space

{\mathcal{H}}_{nm}

as a tensor product of an

n

-dimensional and an

m

-dimensional Hilbert space? We show that the necessary and sufficient conditions can be expressed as a set of linear matrix inequalities (LMIs) on the eigenvalues of

A

M_2\otimes M_2

. This theorem is a consequence of the fact that every density matrix in

M_2\otimes M_m

, with tensor rank smaller or equal to 3, is separable. Although, in

M_3\otimes M_3

, we present an example of SPC matrix with tensor rank 3 that is not PPT. We shall also provide a non trivial example of a family of matrices in

M_k\otimes M_k

, in which both, the SPC and PPT properties, are equivalent. Within this family, there exists a non trivial subfamily in which the SPC property is equivalent to separability.