  The structural study of entanglement in multipartite systems is hindered by
the lack of necessary and sufficient operational criteria able to discriminate
among the various entanglement properties of a given mixed state. Here, we
pursue a different route to the study of multipartite entanglement based on the
closeness of a multipartite state to the set of separable ones. In particular,
we analyze multipartite diagonal symmetric N qubit states and provide the
analytical expression for their Best Separable Approximation (BSA [Phys. Rev.
Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a
separable part and an entangled one with maximal weight of the separable one.
