Einstein Meets Turing: The Computability of Nonlocal Games

Quantum entanglement - the phenomenon where distant particles can be correlated in ways that cannot be explained by classical physics - has mystified scientists since the 1930s, when quantum theory was beginning to emerge. Investigation into fundamental questions about quantum entanglement has continually propelled seismic shifts in our understanding of nature. Examples include Einstein, Podolsky and Rosen's famous 1935 paper about the incompleteness of quantum mechanics, and John Bell's refutation of EPR's argument, 29 years later, via an experiment to demonstrate the non-classicality of quantum entanglement. More recently, the field of quantum computing has motivated researchers to study entanglement in information processing contexts. One question of deep interest concerns the computability of nonlocal games, which are mathematical abstractions of Bell's experiments. The question is simple: is there an algorithm to compute the optimal winning probability of a quantum game - or at least, approximate it? In this paper, I will discuss a remarkable connection between the complexity of nonlocal games and classes in the arithmetical hierarchy. In particular, different versions of the nonlocal games computability problem neatly line up with the problems of deciding Sigma(0)(1), Pi(0)(1), and Pi(0)(2) sentences, respectively.