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arXiv · 2510.04943

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
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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 Navascu\'es-Pironio-Ac\'in (NPA) hierarchy does not attain the commuting operator value at any finite level.

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Marco Fanizza, Larissa Kroell, Arthur Mehta, Connor Paddock, Denis Rochette, William Slofstra, Yuming Zhao. 2025-10-06. The NPA hierarchy does not always attain the commuting operator value. https://arxiv.org/abs/2510.04943

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