A Bell inequality is a constraint on a set of correlations whose violation can be used to certify nonlocality. 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-signaling distributions, we find all 175 Bell inequality classes in the (4,4,2,2) scenario as well as provide a partial list of 18 277 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 21 170 classes in the (3,3,3,3) scenario. Our inequalities are given as Supplemental Material. Finally, we discuss the application of these inequalities to the detection loophole problem and provide lower bounds on the detection efficiency threshold for small numbers of inputs and outputs.

• Received 25 April 2019

DOI:https://doi.org/10.1103/PhysRevA.100.022114