Requirements for copatibility between local and multipartite quantum states

We consider a partial trace transformation which maps a multipartite quantum state to collection of local density matrices. We call this collection a mean field state. For the Hilbert spaces (C2)⊗n and C2 ⊗ C2 ⊗ C4 the necessary and sufficient conditions under which a mean field state is compatible with at least one multipartite pure state are found. Compatibility of mean field states with more general classes of multipartite quantum states is discussed.