Near-Optimal and Explicit Bell Inequality Violations

TL;DR

This paper introduces two new explicit Bell inequality violation games demonstrating stronger quantum correlations than classical strategies, with near-optimal winning probabilities linked to entanglement dimension.

Contribution

The paper presents two novel two-player games based on quantum communication complexity and integrality gaps, providing explicit, stronger Bell inequality violations with near-optimal winning probabilities.

Findings

Quantum strategies significantly outperform classical ones in the new games.

Winning probabilities approach 1 with entanglement, while classical strategies are limited.

The results are nearly optimal in terms of entanglement dimension and output size.

Abstract

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 ${1}/{2}$ 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.