A Noncommutative Nullstellensatz for Perfect Two-Answer Quantum Nonlocal Games

TL;DR

This paper establishes a noncommutative Nullstellensatz related to quantum nonlocal games, proving that perfect quantum strategies imply perfect classical strategies in both finite and infinite-dimensional settings, advancing understanding of quantum-classical correlations.

Contribution

It generalizes the perfect strategy equivalence from finite to infinite-dimensional quantum nonlocal games using a noncommutative Nullstellensatz.

Findings

Perfect quantum strategies imply perfect classical strategies for two-answer games.

Extension of the result to infinite-dimensional quantum strategies.

Introduction of a noncommutative Nullstellensatz related to quantum nonlocality.

Abstract

This paper introduces a noncommutative version of the Nullstellensatz, motivated by the study of quantum nonlocal games. It has been proved that a two-answer nonlocal game with a perfect quantum strategy also admits a perfect classical strategy. We generalize this result to the infinite-dimensional case, showing that a two-answer game with a perfect commuting operator strategy also admits a perfect classical strategy. This result induces a special case of noncommutative Nullstellensatz.