Certifying optimality of Bell inequality violations: noncommutative polynomial optimization through semidefinite programming and local optimization

Hrga, Timotej (author), Klep, Igor (author), Povh, Janez (author)

Bell inequalities are pillars of quantum physics in that their violations imply that certain properties of quantum physics (e.g., entanglement) cannot be represented by any classical picture of physics. In this article Bell inequalities and their violations are considered through the lens of noncommutative polynomial optimization. Optimality of these violations is certified for a large majority of a set of standard Bell inequalities, denoted A2-A89 in the literature. The main techniques used in the paper include the NPA hierarchy, i.e., the noncommutative version of the Lasserre semidefinite programming (SDP) hierarchies based on the Helton-McCullough Positivstellensatz,the Gelfand-Naimark-Segal (GNS) construction with a novel use of the Artin-Wedderburn theory for rounding and projecting, and nonlinear programming (NLP). A new "Newton chip"-like technique for reducing sizes of SDPs arising in the constructed polynomial optimization problems is presented.This technique is based on conditional expectations. Finally, noncommutative Gröbner bases are exploited to certify when an optimizer (a solution yielding optimum violation) cannot be extracted from a dual SDP solution.

• Language: English
• Keywords: noncommutative polynomial, Bell inequality, violation, Gröbner basis, semidefinite programming, eigenvalue optimization, GNS construction, Artin-Wedderburn theory
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FS - Faculty of Mechanical Engineering; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Author Accepted Manuscript
• Year: 2024
• Number of pages: Str. 1341-1373
• Numbering: Vol. 34, iss. 2
• PID: 20.500.12556/RUL-175239
• UDC: 519.8
• ISSN on article: 1052-6234
• DOI: 10.1137/22M1473340
• COBISS.SI-ID: 197627907
• Publication date in RUL: 22.10.2025

Record is a part of a journal

• Title: SIAM journal on optimization
• Shortened title: SIAM j. optim.
• Publisher: SIAM
• ISSN: 1052-6234
• COBISS.SI-ID: 12144985

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• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
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