The NPA hierarchy does not always attain the commuting operator value
Marco Fanizza, Larissa Kroell, Arthur Mehta, Connor Paddock, Denis Rochette, William Slofstra, Yuming Zhao
TL;DR
The paper proves that it is undecidable to determine whether the commuting-operator value of a nonlocal game exceeds 1/2, by constructing a computable reduction from Turing machine halting into BCS nonlocal games through a robust algebraic framework of BCS algebras and nested conjugacy relations. It shows that the NPA hierarchy can fail to attain the commuting-operator value at any finite level for some games, by creating an L-family of Boolean constraint systems whose commuting-operator value is tied to halting. The approach blends approximate representations, GNS construction, and embedding theorems to bridge computability theory with operator-algebraic descriptions of nonlocal games, yielding RE-hardness results and insights into the limits of SDP-based hierarchies in quantum information. This work thus provides a fundamentally algebraic route to undecidability in quantum nonlocality, complementing previous MIP* results and highlighting fundamental limits of hierarchy-based approximations.
Abstract
We show that it is undecidable to determine whether the commuting operator value of a nonlocal game is strictly greater than 1/2. As a corollary, there is a boolean constraint system (BCS) game for which the value of the Navascués-Pironio-Acín (NPA) hierarchy does not attain the commuting operator value at any finite level. Our contribution involves establishing a computable mapping from Turing machines to 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. Our techniques are algebraic and distinct from those used to establish MIP*=RE.