Optimal reducibility of all W states equivalent under stochastic local operations and classical communication

We show that all multipartite pure states that are stochastic local operation and classical communication (SLOCC) equivalent to the N-qubit W state can be uniquely determined (among arbitrary states) from their bipartite marginals. We also prove that only (N - 1) of the bipartite marginals are sufficient and that this is also the optimal number. Thus, contrary to the Greenberger-Horne-Zeilinger (GHZ) class, W-type states preserve their reducibility under SLOCC. We also study the optimal reducibility of some larger classes of states. The generic Dicke states vertical bar GD(N)(l)> are shown to be optimally determined by their (l + 1)-partite marginals. The class of "G" states (superposition of W and (W) over bar) are shown to be optimally determined by just two (N - 2)-partite marginals.