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Geometric Decompositions of Bell Polytopes with Practical Applications



Peter L. Bierhorst


In the well-studied (2,2,2) Bell experiment consisting of two parties, two measurement settings per party, and two possible outcomes per setting, it is known that if the experiment obeys no- signaling constraints, then the set of admissible experimental probability distributions is fully characterized as the convex hull of 24 distributions: 8 Popescu-Rohrlich (PR) boxes and 16 local deterministic distributions. Here, we refine this result to show that any nonlocal nonsignaling distribution can always be uniquely expressed as a convex combination of exactly one PR box and (up to) eight local deterministic distributions. In this representation each PR box will always occur only with a fixed set of eight local deterministic distributions with which it is affiliated. This decomposition has multiple applications: we demonstrate an analytical proof that the minimum detection efficiency for which nonlocality can be observed is eta>2/3 even for theories constrained only by the no-signaling principle, and we develop new algorithms that speed the calculation of important statistical functions of Bell test data. Finally, we enumerate the vertices of the no-signaling polytope for the (2, n, 2) ``chained Bell'' scenario and find that similar decomposition results are possible in this general case. Here, our results allow us to prove the optimality of a bound, derived in [Barrett et al. 2006, Phys. Rev. Lett. 97, 170409] on the proportion of local theories in a local/nonlocal mixture that can be inferred from the experimental violation of a chained Bell inequality.
Journal of Physics A: Mathematical and Theoretical


Bell’s inequality, No-signaling polytopes, Quantum nonlocality


Bierhorst, P. (2016), Geometric Decompositions of Bell Polytopes with Practical Applications, Journal of Physics A: Mathematical and Theoretical, [online], (Accessed December 9, 2023)
Created April 20, 2016, Updated May 19, 2020