# Connes' Embedding Problem and Winning Strategies for Quantum XOR Games

## TL;DR

This paper links Connes' embedding problem to quantum XOR games, showing that the problem's resolution depends on whether strategies in the commuting model can be approximated in finite dimensions.

## Contribution

It establishes an equivalence between Connes' embedding problem and the existence of finite-dimensional strategies for all quantum XOR games.

## Key findings

- Connes' embedding problem is equivalent to the equality of entanglement and commuting biases in quantum XOR games.
- The problem reduces to whether quantum XOR games with commuting strategies also have finite-dimensional strategies.
- Provides a new perspective connecting operator algebra problems with quantum game strategies.

## Abstract

We consider quantum XOR games, defined in [11], from the perspective of unitary correlations defined in [7]. We show that Connes' embedding problem has a positive answer if and only if every quantum XOR game has entanglement bias equal to the commuting bias. In particular, the embedding problem is equivalent to determining whether every quantum XOR game $G$ with a winning strategy in the commuting model also has a winning strategy in the approximate finite-dimensional model.