The NPA hierarchy does not always attain the commuting operator value

2025-10-06

AI Summary

This paper investigates the decidability of the commuting-operator value in nonlocal games. Specifically, it addresses the decision problem of determining whether this value strictly exceeds 1/2. The authors construct a computable reduction from the Halting Problem to Boolean Constraint System (BCS) games, thereby establishing, for the first time, the undecidability of this task. Furthermore, they explicitly construct a BCS game whose commuting-operator value cannot be approximated at any finite level of the Navascués–Pironio–Acín (NPA) hierarchy—demonstrating its failure to converge. These results rely solely on algebraic coding, operator-algebraic techniques, and nonlocal-game modeling, and are fully independent of the MIP* = RE framework. The work thus provides a foundational counterexample concerning both the computational complexity of commuting-operator values and the limitations of hierarchical approximation schemes in quantum information theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Imperfect Information
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• The NPA hierarchy fails to achieve commuting operator values in finite levels
• Determining if commuting operator value exceeds 1/2 is undecidable
• Halting problem is encoded into nonlocal game values via computable mapping

Innovation

• Mapping Turing machines to BCS nonlocal games computably
• Encoding halting problems as commuting operator decisions
• Using algebraic techniques distinct from MIP*=RE methods