  Inspired by the `computable cross norm' or `realignment' criterion, we
propose a new point of view about the characterization of the states of
bipartite quantum systems. We consider a Schmidt decomposition of a bipartite
density operator. The corresponding Schmidt coefficients, or the associated
symmetric polynomials, are regarded as quantities 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 strengthened in such a way to
obtain a new family of criteria that are independent of the original one. This
conjecture is supported by numerical examples for the low dimensional cases.
These ideas can 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.
