Separability criterion for multipartite quantum states based on the Bloch representation of density matrices

We give a new separability criterion, a necessary condition for separability of N partite quantum states. The criterion is based on the Bloch representation of a N -partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N -partite quantum states in H = H1 ⊗H2 ⊗ · · ·⊗HN . The criterion can test whether a N -partite state is entangled and can be applied to different partitions of the N -partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a sufficiency condition for separability of a 3-partite state, straightforwardly generalizable to the case N > 3, under certain condition. We also give a necessary and sufficient condition for separability of a class of N -qubit states which includes N -qubit PPT states.