A necessary and sufficient criterion for multipartite separable states

We present a necessary and sufficient condition for the separability of multipartite quantum states, this criterion also tells us how to write a multipartite separable state as a convex sum of separable pure states. To work out this criterion, we need to solve a set of equations, actually it is easy to solve these quations analytically if the density matrix of the given quantum state has few nonzero eigenvalues.