  We consider the Schmidt decomposition of a bipartite density operator induced
by the Hilbert-Schmidt scalar product, and we study the relation between the
Schmidt coefficients and entanglement. First, we define the Schmidt equivalence
classes of bipartite states. Each class consists of all the density operators
(in a given bipartite Hilbert space) sharing the same set of Schmidt
coefficients. Next, we review the role played by the Schmidt coefficients with
respect to the separability criterion known as the `realignment' or `computable
cross norm' criterion; in particular, we highlight the fact that this criterion
relies only on the Schmidt equivalence class of a state. Then, the relevance --
with regard to the characterization of entanglement -- of the `symmetric
polynomials' in the Schmidt coefficients and a new family of separability
criteria that generalize the realignment criterion are discussed. Various
interesting open problems are proposed.
