The NPA hierarchy does not always attain the commuting operator value
Quantum Physics
2025-12-19 v4 Computational Complexity
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.
Cite
@article{arxiv.2510.04943,
title = {The NPA hierarchy does not always attain the commuting operator value},
author = {Marco Fanizza and Larissa Kroell and Arthur Mehta and Connor Paddock and Denis Rochette and William Slofstra and Yuming Zhao},
journal= {arXiv preprint arXiv:2510.04943},
year = {2025}
}
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45 pages