Volume 8 (2012) Article 27 pp. 623-645
Near-Optimal and Explicit Bell Inequality Violations
Revised: August 31, 2012
Published: December 27, 2012
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Keywords: quantum communication, nonlocal games, Bell inequalities, Fourier analysis
ACM Classification: J.2
AMS Classification: 81P65

Abstract: [Plain Text Version]

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., $\log n$ EPR-pairs), while we show that the winning probability of any classical strategy differs from $\frac{1}{2}$ by at most $O((\log n)/\sqrt{n})$.

The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here $n$-dimensional entanglement allows the game to be won with probability $1/(\log n)^2$, while the best winning probability without entanglement is $1/n$. This near-linear ratio is almost optimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players.

An earlier version of this paper appeared in the Proceedings of the 26th IEEE Conference on Computational Complexity, pages 157--166, 2011.