Near-Optimal and Explicit Bell Inequality Violations

Entangled quantum systems can exhibit correlations that cannot be simulated classically. For historical reasons such correlations are called “Bell inequality violations.” We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a strategy using a maximally entangled state with local dimension n (e. g., logn EPR-pairs), while we show that the winning probability of any classical strategy differs from 1/2 by at most O((logn)/ √ n). ∗An earlier version of this paper appeared in the Proceedings of the 26th IEEE Conference on Computational Complexity, pages 157–166, 2011. †Supported by a Vici grant from the Netherlands Organisation for Scientific Research (NWO), and by the European Commission under the project QCS (Grant No. 255961). ‡Supported by the Israel Science Foundation, by the Wolfson Family Charitable Trust, and by a European Research Council (ERC) Starting Grant. Part of the work done while a DIGITEO visitor in LRI, Orsay. §Supported by a Vidi grant from the Netherlands Organisation for Scientific Research (NWO), and by the European Commission under the project QCS (Grant No. 255961). ¶Supported by a Vidi grant from the Netherlands Organisation for Scientific Research (NWO), and by the European Commission under the project QCS (Grant No. 255961). ACM Classification: J.2 AMS Classification: 81P68