Separability for Weak Irreducible matrices

This paper is devoted to the study of the separability problem in the field of Quantum information theory. We deal mainly with the bipartite finite dimensional case and with two types of matrices, one of them being the PPT matrices. We proved that many results holds for both types. If these matrices have specific Hermitian Schmidt decompositions then the matrices are separable in a very strong sense. We proved that both types have what we call split decompositions. We defined the notion of weak irreducible matrix, based on the concept of irreducible state defined recently. These split decomposition theorems together with the notion of weak irreducible matrix, imply that these matrices are weak irreducible or a sum of weak irreducible matrices of the same type. The separability problem for these types of matrices can be reduced to the set of weak irreducible matrices of the same type. We also provided a complete description of weak irreducible matrices of both types. Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable, we found sharp inequalites providing separability for both types.