Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy

TL;DR

This paper explores the computational complexity of determining exact quantum values of nonlocal games, establishing their classification within the arithmetical hierarchy and introducing new compression theorems.

Contribution

It proves that deciding whether the quantum value of a nonlocal game is exactly 1 is $oldsymbol{ ext{Pi}_2}$-complete, advancing understanding of the complexity of quantum game evaluation.

Findings

Exact quantum value decision is $oldsymbol{ ext{Pi}_2}$-complete.

Introduces a new gapless compression theorem for quantum strategies.

Provides an alternative proof that the set of quantum correlations is not closed.

Abstract

We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is $1$ or at most $\frac{1}{2}$ is complete for the class $\Sigma_1$ (i.e., $\mathsf{RE}$). A result of Slofstra implies that deciding whether the commuting operator value of a nonlocal game is equal to $1$ is complete for the class $\Pi_1$ (i.e., $\mathsf{coRE}$). We prove that deciding whether the quantum value of a two-player nonlocal game is exactly equal to $1$ is complete for $\Pi_2$; this class is in the second level of the arithmetical hierarchy and corresponds to formulas of the form "$\forall x \, \exists y \, \phi(x,y)$". This shows that exactly computing the quantum value is strictly harder than approximating it, and also strictly harder than computing the commuting operator value (either exactly or approximately). We explain how results about the complexity of nonlocal games all follow in a unified manner from a technique known as compression. At the core of our $\Pi_2$-completeness result is a new "gapless" compression theorem that holds for both quantum and commuting operator strategies. Our compression theorem yields as a byproduct an alternative proof of Slofstra's result that the set of quantum correlations is not closed. We also show how a "gap-preserving" compression theorem for commuting operator strategies would imply that approximating the commuting operator value is complete for $\Pi_1$.