Bell Inequalities From No-Signalling Distributions

Abstract

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.

arXiv comment15 + 7 pages. v2: more scenarios are covered and more analysis has been done. v3: shorter title and a few additional updates, including summary table 1

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Cited by in corpus (19)

• Photonic verification of device-independent quantum key distribution against collective attacks
• Device-independent quantum key distribution with random postselection
• Maximal Violation of a Broad Class of Bell Inequalities and Its Implications on Self-Testing
• Equivalence between face nonsignaling correlations, full nonlocality, all-versus-nothing proofs, and pseudotelepathy
• Towards correlation self-testing of quantum theory in the adaptive Clauser-Horne-Shimony-Holt game
• Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms
• Experimentally friendly approach towards nonlocal correlations in multisetting N -partite Bell scenarios
• Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles
• Analysing causal structures using Tsallis entropies
• Revisited aspects of the local set in CHSH Bell scenario
• Self-testing tilted strategies for maximal loophole-free nonlocality
• On the insufficiency of entropic inequalities for detecting non-classicality in the Bell causal structure
• Class of Bell-Clauser-Horne inequalities for testing quantum nonlocality
• Experimental single-copy distillation of quantumness from higher-dimensional entanglement
• Witnessing Bell violations through probabilistic negativity
• Tight Bell inequalities from polytope slices
• The Binary-Outcome Detection Loophole
• How Likely Are You to Observe Non-locality with Imperfect Detection Efficiency and Random Measurement Settings?
• Trading symmetry for Hilbert-space dimension in Bell-inequality violation