Quantum XOR Games

TL;DR

Quantum XOR games extend classical XOR games by incorporating quantum states as questions, revealing new behaviors and advantages of entanglement, and providing efficient approximation algorithms for optimal performance.

Contribution

The paper introduces quantum XOR games, demonstrating their unique properties and developing an efficient approximation algorithm for their optimal strategies.

Findings

Entanglement can lead to arbitrarily large advantages in quantum XOR games.

Sharing a maximally entangled state offers only a small advantage over no entanglement.

An efficient algorithm approximates the best performance in quantum XOR games.

Abstract

We introduce quantum XOR games, a model of two-player one-round games that extends the model of XOR games by allowing the referee's questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck's inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance players can obtain in a given game, both in case they have no shared entanglement and in case they share unlimited entanglement. As a byproduct of the algorithm we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all.