The structure of optimal and nearly-optimal quantum strategies for non-local XOR games

TL;DR

This paper characterizes the structure of optimal and nearly-optimal quantum strategies for non-local XOR games, including a broad class of CHSH(n) games, using relations and representation theory tools.

Contribution

It establishes a general framework linking quantum strategies to specific relations and extends analysis to the CHSH(n) family of XOR games.

Findings

Quantum strategies are characterized by specific relations.

Optimal strategies satisfy these relations exactly.

Nearly-optimal strategies approximately satisfy the relations.

Abstract

We study optimal and nearly-optimal quantum strategies for non-local XOR games. First, we prove the following general result: for every non-local XOR game, there exists a set of relations with the properties: (1) a quantum strategy is optimal for the game if and only if it satisfies the relations, and (2) a quantum strategy is nearly optimal for the game if and only if it approximately satisfies the relations. Next, we focus attention on a specific infinite family of XOR games: the CHSH(n) games. This family generalizes the well-known CHSH game. We describe the general form of CHSH(n) optimal strategies. Then, we adapt the concept of intertwining operator from representation theory and use that to characterize nearly-optimal CHSH(n) strategies.