Bipartite quantum systems: on the realignment criterion and beyond

Inspired by the computable cross norm, or realignment criterion, we present a new point of view about the characterization of the states of bipartite quantum systems. We consider the Schmidt decomposition of a bipartite density operator with respect of the HilbertSchmidt product. The corresponding Schmidt coefficients, or the associated symmetric polynomials, are discussed as functions that can be used to characterize bipartite quantum states. In particular, starting from the realignment criterion, a family of necessary conditions for the separability of bipartite quantum states is derived. We conjecture that these conditions, which are weaker than the parent criterion, can be extended to a new family of criteria which are independent of the original one. This is confirmed by numerical examples for the low dimensional cases. These ideas can as well be applied to the study of quantum channels, leading to a relation between the rate of contraction of a map and its ability to preserve entanglement. PACS numbers: 03.67.-a, 03.67.Mn