Abstract
Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. 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 quantum 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 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 to win the game with probability 1/(log n)^2, while the best winning probability without entanglement is 1/n. This near-linear ratio (``Bell inequality violation'') is near-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.
Highlights
One of the most striking features of quantum mechanics is the fact that entangled particles can exhibit correlations that cannot be reproduced or explained by classical physics, or more precisely, by “local hidden-variable theories.”
Experimental realization of such correlations is the strongest proof we have that nature does not behave according to classical physics: nature cannot simultaneously be “local” and “realistic”
All behave in accordance with the predictions of quantum mechanics, though so far none has closed all “loopholes” that would allow some classical explanation of the observations based on imperfect behavior of, for instance, the photon detectors used
Summary
One of the most striking features of quantum mechanics is the fact that entangled particles can exhibit correlations that cannot be reproduced or explained by classical physics, or more precisely, by “local hidden-variable theories.” This was first noted by Bell [2] in response to Einstein, Podolsky, and Rosen’s challenge to the completeness of quantum mechanics [9]. NEAR-OPTIMAL AND EXPLICIT BELL INEQUALITY VIOLATIONS the inputs x ∈ {0, 1} and y ∈ {0, 1} are uniformly distributed, and Alice and Bob win the game if their respective outputs a ∈ {0, 1} and b ∈ {0, 1} satisfy a ⊕ b = x ∧ y; in other words, a should equal b unless x = y = 1. In terms of upper bounds, [14] proved that the maximum Bell inequality violation ωn∗(G)/ω(G) obtainable with entangled strategies of local dimension n, is at most O(n), and [13, Theorem 6.8] proved that if Alice and Bob have at most k possible outputs each, the violation ω∗(G)/ω(G) is at most O(k), irrespective of the amount of entanglement they can use. In the remainder of this introduction we provide an overview of the two games, followed by some discussion and comparison
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