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 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.

Figure 1

Plots of the critical lines $Z_2$, $Z_4$, and $Z_{\infty }$. The witness obtained from the Bell inequality with 4 settings already provides a significant improvement over the case of 2 settings. The black point in the inset shows the data point from Ref. [9], with $N=476\pm 21$.

Figure 2

Upper bound on the value of ${\zeta }_a^2$ required to see a violation of the Bell correlation witness (18). The bound depends on the number of particles $N$.

Figure 3

Number of experimental runs per spins required to rule out non-Bell-correlated states with a confidence of $1-ϵ$ as a function of ${\mathcal{C}}_b$ and ${\zeta }_a$. For ${\mathcal{C}}_b=0.98$ and ${\zeta }_a^2=0.272$ (as reported in Ref. [9]), approximately $17\ln (100)\simeq 80$ runs per spin are sufficient to reach a confidence level of 99%.