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

TL;DR

This paper demonstrates that all states SLOCC equivalent to the N-qubit W state can be uniquely identified from a minimal set of bipartite marginals, highlighting their optimal reducibility and contrasting with GHZ states.

Contribution

It establishes the minimal bipartite marginals needed to uniquely determine W-class states and extends the analysis to Dicke and G states, showing their optimal reducibility.

Findings

W states are uniquely determined by (N-1) bipartite marginals.

Dicke states are determined by (ell+1)-partite marginals.

G states are determined by two (N-2)-partite marginals.

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.