Optimal reducibility of all Stochastic Local Operation and Classical Communication equivalent W states

Abstract: We show that all multipartite pure states that are 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 this is also the optimal number. Thus, contrary to the $GHZ$ 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 $|GD_N^\ell\ran$ are shown to be optimally determined by their ($\ell+1$)-partite marginals. The class of `$G$' states (superposition of $W$ and $\bar{W}$) are shown to be optimally determined by just two $(N-2)$-partite marginals.