Partial trace inequalities for partial transpose of positive semidefinite block matrices

Daeshik Choi

We present inequalities related to partial transpose and partial traces of a positive semidefinite matrix.

Daeshik Choi

Mohammad Bagher Ghaemi, Nahid Gharakhanlu, Themistocles M. Rassias +1 more

Christopher King

Yixuan Liang, Jiahao Yan, Dongran Si +1 more

Abstract We show that the partial transpose of 9 × 9 positive semidefinite matrices do not have inertia ( 4 , 1 , 4 ) and ( 3 , 2 , 4 ) . It solves an open problem in ‘LINEAR AND MULTILINEAR ALGEBRA. Changchun Feng et al, 2022’. We apply our results to construct some inertia, as well as present the list of all possible inertia of partial transpose of 12 × 12 positive semidefinite 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.