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 to characterize correlations in many-body systems. As an example, we illustrate our method with the experimental data presented in [Science 352, 441 (2016)]