Norm Conditions for Separability in ${\mathbb M}_m\otimes {\mathbb M}_n$

Abstract

An element $\mathbf{S}$ of the tensor product ${\mathbb M}_m\otimes {\mathbb M}_n$ is said to be separable if it admits a (separable) decomposition

$$ \mathbf{S}\ =\ \sum _pX_p\otimes Y_p \quad \exists \ \ 0 \le X_p \in {\mathbb M}_m,\ \exists \ 0 \le Y_p \in {\mathbb M}_n. $$

This decomposition is not unique. We present some conditions on suitable norms of $\mathbf{S}$ which guarantee its separability. Even when separability of $\mathbf{S}$ is guaranteed by some method, its separable decomposition itself is difficult to construct. We present a general condition which makes it possible to find a way of an explicit separable decomposition.