The perturbative method for quantum correlations

Quantum Physics [Submitted on 27 Mar 2026] The perturbative method for quantum correlations Sacha Cerf, Harold Ollivier The set \mathcal{Q} of quantum correlations is the collection of all possible probability distributions on measurement outcomes achievable by space-like separated parties sharing a quantum state. Since the original work of Tsirelson [Tsirelson, Lett. Math. Phys. 4, 93 (1980)], this set has mainly been studied through the means algebraic and convex geometry techniques. We introduce a perturbative method using Lie-theoretic tools for the unitary group to analyze the response of the evaluations of Bell functionals under infinitesimal unitary perturbations of quantum strategies. Our main result shows that, near classical deterministic points, an (n, 2, d) Bell operator decomposes into a direct sum of (k, 2, d-1) Bell operators which we call \emph{subset games}. We then derive three key insights: (1) in the (n, 2, 2) case, if p_0 is classically optimal, it remains locally optimal even among 2-dimensional quantum strategies, implying in turn that the boundary of \mathcal{Q} is flat around classical deterministic points; (2) it suggests a proof strategy for Gisin's open problem on correlations in \mathcal{Q}(D) unattainable by projective strategies of the same dimension; and (3) it establishes that the Ansatz dimension is a critical resource for learning in distributed scenarios, even when the optimal solution admits a low-dimensional representation. Comments: 15 pages + 3 pages of appendix, 2 figures Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2603.26875 [quant-ph]   (or arXiv:2603.26875v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.26875 Focus to learn more Submission history From: Sacha Cerf [view email] [v1] Fri, 27 Mar 2026 18:00:08 UTC (686 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)