Optimal reducibility of all W states equivalent under stochastic local operations and classical communication
Swapan Rana, Preeti Parashar · Physical Review A · 2011
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 $(N\ensuremath{-}1)$ 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 $|{\text{GD}}_{N}^{\ensuremath{\ell}}\ensuremath{\rangle}$ are shown to be optimally determined by their ($\ensuremath{\ell}+1$)-partite marginals. The class of ``$G$'' states (superposition of $W$ and $\overline{W}$) are shown to be optimally determined by just two $(N\ensuremath{-}2)$-partite marginals.