Necessary and sufficient conditions for local manipulation of multipartite pure quantum states

Suppose that several parties jointly possess a pure multipartite state, vertical bar psi >. Using local operations on their respective systems and classical communication (i.e. LOCC), it may be possible for the parties to transform deterministically vertical bar psi > into another joint state vertical bar phi >. In the bipartite case, the Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states are in the same stochastic LOCC (SLOCC) class. Here, we generalize the Nielsen majorization theorem to the multipartite case, and find necessary and sufficient conditions for the existence of a local separable transformation between two multipartite states in the same SLOCC class. When such a deterministic conversion is not possible, we find an expression for the maximum probability to convert one state to another by local separable operations. In addition, we find necessary and sufficient conditions for the existence of a separable transformation that converts a multipartite pure state into one of a set of possible final states all in the same SLOCC class. Our results are expressed in terms of (i) the stabilizer group of the state representing the SLOCC orbit and (ii) the associate density matrices (ADMs) of the two multipartite states. The ADMs play a similar role to that of the reduced density matrices when considering local transformations that involve pure bipartite states. We show, in particular, that the requirement that one ADM majorizes another is a necessary condition but is, in general, far from also being sufficient as it happens in the bipartite case. In most of the results the twirling operation with respect to the stabilizer group (of the representative state in the SLOCC orbit) plays an important role that provides a deep link between entanglement theory and the resource theory of reference frames.