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

An element $$\mathbf{S}$$ S of the tensor product $${\mathbb M}_m\otimes {\mathbb M}_n$$ M m ⊗ M n is said to be separable if it admits a (separable) decomposition $$ \mathbf{S}\ =\ \sum _pX_p\otimes Y_p \quad \exists \ \ 0 \le X_p \in {\mathbb M}_m,\ \exists \ 0 \le Y_p \in {\mathbb M}_n. $$ S = ∑ p X p ⊗ Y p ∃ 0 ≤ X p ∈ M m , ∃ 0 ≤ Y p ∈ M n . This decomposition is not unique. We present some conditions on suitable norms of $$\mathbf{S}$$ S which guarantee its separability. Even when separability of $$\mathbf{S}$$ 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.