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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·October 6, 2025
Quantum PhysicsComplexity
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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 Navascués-Pironio-Acín (NPA) hierarchy does not attain the commuting operator value at any finite level.