Optimal reducibility of allWstates equivalent under stochastic local operations and classical communication

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 are shown to be optimally determined by their ( + 1)-partite marginals. The class of 'G' states (superposition of W andW ) are shown to be optimally determined by just two (N − 2)-partite marginals.

Determination of stabilizer states

“…According to the Schwarz inequality, we have a iω s iω s = (−1) f (iω s iω s )+f (iω s iω s +rs) • a iω s iω s +rs , (6) where i ωs , i ωs are two arbitrary vectors belong to space…”

“…They also shown that the set of just about half the parties contains all the information in almost all n-party pure states [4]. In recent years much effort has been spent on studying quantum states with special forms and properties, such as stochastic local operations and classical communication (SLOCC) equivalent |W n states [5][6][7], Dicke states [8], stabilizer states [9] and generalized |GHZ n states which are LU equivalent to some stabilizer states [10,11]. In [7], the present authors proved that SLOCC equivalent * Email: gaofei bupt@hotmail.com |W n states can be uniquely determined among arbitrary states by their (n − 1) bipartite reduced density matrices whose index set is correspond to a tree graph.…”

Trade-off relation among genuine three-qubit nonlocalities in four-qubit systems

“…Nonlocal correlations are proposed in three-qubit generalized GHZ states and four-qubit generalized GHZ states [23]. Meanwhile, all multipartite pure states that are equivalent to the N-qubit W states under stochastic local operation and classical communication (SLOCC) can be uniquely determined (among arbitrary states) from their bipartite marginals [24]. Two overlapping bipartite binary Bell inequalities cannot be simultaneously violated, which would contradict the usual no-signaling principle.…”