Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy*

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 1/2 is complete for the class Sigma(1) (i.e., 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., 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 "for all x there 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. All previous works only study the setting of finite-dimensional quantum strategies; ours is the first to study compression of games in the commuting operator setting. 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.