Bell correlations in a many-body system with finite statistics

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 paper, 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.