Bounding the Set of Classical Correlations of a Many-Body System

We present a method to certify the presence of Bell correlations in experimentally observed statistics, and to obtain new Bell inequalities. Our approach is based on relaxing the conditions defining the set of correlations obeying a local hidden variable model, yielding a convergent hierarchy of semidefinite programs (SDP’s). Because the size of these SDP’s is independent of the number of parties involved, this technique allows us to characterize correlations in many-body systems. As an example, we illustrate our method with the experimental data presented in Science 352, 441 (2016).

Figure 1

For $N=10$ and $K=2$, the plane of the symmetric correlations of the form $\alpha {\overrightarrow{\mathcal{S}}}_2^{(1)}+\beta {\overrightarrow{\mathcal{S}}}_2^{(2)}$, with ${\overrightarrow{\mathcal{S}}}_2^{(1)}=(1,-1,0,-1,1{)}^T/\sqrt{4}$ and ${\overrightarrow{\mathcal{S}}}_2^{(2)}=(0,-1,-1,1,0{)}^T/\sqrt{3}$. In blue, the intersection of ${\mathbb{P}}_2^{\mathrm{S}}$ with the plane, computed with a linear program. In red, the boundary of the feasible set of SDP (3) for $\mu =1$. The gap between the two objects is imputable mainly to the first relaxation, and the small $N$ was chosen also to appreciate its size, which remains of the same order while ${\mathbb{P}}_2^{\mathrm{S}}$ increases with $N$ (see also Fig. 2).

Figure 2

Plane generated by $\{{\mathcal{S}}_0,({\mathcal{S}}_{00}+2{\mathcal{S}}_{01}+{\mathcal{S}}_{11})\}$. Black circled dot, point $(367.6,-525.4)$ measured experimentally in [18] for $N=476$. Blue points, projected vertices of ${\mathbb{P}}_2^{\mathrm{S}}$. Blue line, bound given by the Bell inequality $-2{\mathcal{S}}_0+({\mathcal{S}}_{00}+2{\mathcal{S}}_{01}+{\mathcal{S}}_{11})/2+2N\ge 0$, from Refs. [13, 18]. This inequality is tight, meaning that it is also a facet of the projected polytope. Pink region, points where SDP (15) gives $\lambda \ge 1$. Orange dashed line, Bell inequality obtained numerically by solving the dual of SDP (15). The distance between the blue and the orange lines is 1.000 002, meaning that the error of our method compared to the tight classical bound scales as $1/N$, and it is imputable mainly to the first relaxation.