Separability Properties for a Class of Block Matrices

TL;DR

This paper establishes a criterion linking the separability of certain block matrices with their semi-positive definiteness, providing explicit decompositions based on eigenvalues and eigenvectors.

Contribution

It introduces a new equivalence between separability and semi-positive definiteness for block matrices with commuting, normal blocks, along with explicit decomposition methods.

Findings

Separability is equivalent to semi-positive definiteness for the considered block matrices.

A separability decomposition of length equal to the matrix dimension is provided.

Semi-positive definiteness of the block matrix is equivalent to that of smaller matrices.

Abstract

It is shown that, for the block matrices belonging to $M(nd,\mathbb{C})$ with commuting and normal block entries of dimension $d$, the separability of such a block matrices is equivalent to its semi-positive definity. The separability decomposition of lenght equal to the dimension of the block matrix (which is smaller then Carath\'eodory theorem implies) is given. The separability decomposition depends only on eigenvalues of block entries in the first part and on eigenvectors of the block entries in the second part of the tensor product. It is shown that semi-positive definity of considered block matrices is equivalent to semi-positive definity $d$ smaller matrices of dimension $n$.