The NPA hierarchy does not always attain the commuting operator value

Abstract

We show that it is undecidable to determine whether the commuting operator value of a nonlocal game is strictly greater than 1/2. Specifically, there is a computable mapping from Turing machines to /boolean constraint system (BCS) nonlocal games in which the halting property of the machine is encoded as a decision problem for the commuting operator value of the game. As a corollary, there is a BCS game for which the value of the Navascu\'es-Pironio-Ac\'in (NPA) hierarchy does not attain the commuting operator value at any finite level.