Bell Inequalities From No-Signalling Distributions

A Bell inequality is a constraint on a set of correlations whose violation can be used to certify non-locality. They are instrumental for device-independent tasks such as key distribution or randomness expansion. In this work we consider bipartite Bell inequalities where two parties have $m_A$ and $m_B$ possible inputs and give $n_A$ and $n_B$ possible outputs, referring to this as the $(m_A, m_B, n_A, n_B)$ scenario. By exploiting knowledge of the set of extremal no-signalling distributions, we find all 175 Bell inequality classes in the (4, 4, 2, 2) scenario, as well as providing a partial list of 18277 classes in the (4, 5, 2, 2) scenario. We also use a probabilistic algorithm to obtain 5 classes of inequality in the (2, 3, 3, 2) scenario, which we confirmed to be complete, 25 classes in the (3, 3, 2, 3) scenario, and a partial list of 21170 classes in the (3, 3, 3, 3) scenario. Our inequalities are given in supplementary files. Finally, we discuss the application of these inequalities to the detection loophole problem, and provide new lower bounds on the detection efficiency threshold for small numbers of inputs and outputs.