Multipartite classical states and detecting quantum discord

We study various types of multipartite separable states in terms of their inherent classical features. For the two important classes of pseudo-classical, introduced here, and classical states, we provide necessary and sufficient conditions for deciding membership in both which can be checked in polynomial running time. Geometrically, the volume of these states in multipartite state space is found to be measure zero. We also provide a physical criterion for detecting non-classical states based on the commutivity of reduced states following local POVMs performed on individual subsystems.