Separability problem for multipartite states of rank at most four

One of the most important problems in quantum information is the separability problem, which asks whether a given quantum state is separable. We investigate multipartite states of rank at most four which are PPT (i.e., all their partial transposes are positive semidefinite). We show that any PPT state of rank two or three is separable and has length at most four. For separable states of rank four, we show that they have length at most six. It is six only for some qubit-qutrit or multiqubit states. It turns out that any PPT entangled state of rank four is necessarily supported on a 3x3 or a 2x2x2 subsystem. We obtain a very simple criterion for the separability problem of the PPT states of rank at most four: such a state is entangled if and only if its range contains no product vectors. This criterion can be easily applied since a four-dimensional subspace in the 3x3 or 2x2x2 system contains a product vector if and only if its Pluecker coordinates satisfy a homogeneous polynomial equation (the Chow form of the corresponding Segre variety). We have computed an explicit determinantal expression for the Chow form in the former case, while such expression was already known in the latter case.Comment: 19 page