Bell Correlations in a Many-Body System with Finite Statistics

Abstract

A recent experiment reported the first violation of a Bell correlation witness in a many-body system [Science 352, 441 (2016)]. Following discussions in this Letter, we address here the question of the statistics required to witness Bell correlated states, i.e., states violating a Bell inequality, in such experiments. We start by deriving multipartite Bell inequalities involving an arbitrary number of measurement settings, two outcomes per party and one-and two-body correlators only. Based on these inequalities, we then build up improved witnesses able to detect Bell correlated states in many-body systems using two collective measurements only. These witnesses can potentially detect Bell correlations in states with an arbitrarily low amount of spin squeezing. We then establish an upper bound on the statistics needed to convincingly conclude that a measured state is Bell correlated.

Bipartite nonlocality with a many-body system

“…Recently, such a reduction in the number of measurements, applicable when assuming that the measurements are known and trusted, was used to transform a Bell inequality involving an unbounded number of settings into a witness with only two global measurements [3]. It was also used to show that the Svetlichny inequality [22], which is a Bell inequality for N parties requiring the measurement of 2 N correlators, reduces to a negativity condition on the mean value of an observable that can be evaluated with 2 measurement settings only [7].…”

“…Exceptions have been found recently. In particular [2][3][4], showed that some multi-partite states are able to violate Bell inequalities using the product of two local observables. These inequalities paved the way for Bell tests with many-body systems where few-body correlations-correlations between a small number of local observables-are accessible only.…”

“…This requirement have been lifted with the use of a Bell correlation witness (see below for details) which were used to demonstrate the presence of Bell correlations in many-body states [5][6][7]. This implies that if one could measure the constituent particle individually, then the observed correlations are strong enough to violate Bell inequalities presented in [2][3][4]. Such an individual addressing is unrealistic for a large number of particles.…”

Does large quantum Fisher information imply Bell correlations?

“…It turns out that these results can be improved by considering a multi-setting generalization of the Bell inequality (8) presented in Ref. [28]. Again, this inequality can be written as a Bell correlation witness which requires collective spin measurements only.…”

“…First, we take its elements to start with an ansatz for a Bell inequality, which turns out to be of the form of multipartite Bell inequalities based on two-body correlators recently introduced by Tura et al [27]. Considering a multi-setting extension of this Bell inequality [28], we show that (i) only quantum states with F(ρ, A) > N (i.e. beating the standard limit of separable states) can potentially violate the inequality, and (ii) any quantum state with F(ρ, A) > 3N will violate the Bell inequality.…”