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This research investigates the fundamental trade-off between symmetry constraints and Hilbert-space dimension when maximizing violations of Bell inequalities in quantum strategies. The study demonstrates that for certain asymmetric Bell inequalities, maximal violation requires abandoning party-exchange symmetry to utilize minimal-dimensional quantum systems, whereas symmetric correlations suffice for others. These findings refine our understanding of the geometry of quantum correlations and offer new insights into self-testing protocols within foundational quantum information theory.
Trading symmetry for Hilbert-space dimension in Bell-inequality violation
arXiv:2601.02893v3 Announce Type: replace Abstract: In quantum information, asymmetry, i.e., the lack of symmetry, is a resource allowing one to accomplish certain tasks that are otherwise impossible. Similarly, in a Bell test using any given Bell inequality, the maximum violation achievable using quantum strategies respecting or disregarding a certain symmetry can be different. In this work, we focus on the symmetry involved in the exchange of parties and explore when we have to trade this symmetry for a lower-dimensional quantum strategy in achieving the maximal violation of given Bell inequalities. For the family of symmetric Collins-Gisin-Linden-Massar-Popescu inequalities, we provide evidence showing that there is no such trade-off. However, for several other Bell inequalities with a small number of dichotomic measurement settings, we show that symmetric quantum strategies in the minimal Hilbert space dimension can only lead to a suboptimal Bell violation. In other words, there exist symmetric Bell inequalities that can only be maximally violated by asymmetric quantum strategies of minimal dimension. In contrast, one can also find examples of asymmetric Bell inequalities that are maximally violated by symmetric correlations. The implications of these findings on the geometry of the set of quantum correlations and the possibility of performing self-testing therefrom are briefly discussed.
Abstract:In quantum information, asymmetry, i.e., the lack of symmetry, is a resource allowing one to accomplish certain tasks that are otherwise impossible. Similarly, in a Bell test using any given Bell inequality, the maximum violation achievable using quantum strategies respecting or disregarding a certain symmetry can be different. In this work, we focus on the symmetry involved in the exchange of parties and explore when we have to trade this symmetry for a lower-dimensional quantum strategy in achieving the maximal violation of given Bell inequalities. For the family of symmetric Collins-Gisin-Linden-Massar-Popescu inequalities, we provide evidence showing that there is no such trade-off. However, for several other Bell inequalities with a small number of dichotomic measurement settings, we show that symmetric quantum strategies in the minimal Hilbert space dimension can only lead to a suboptimal Bell violation. In other words, there exist symmetric Bell inequalities that can only be maximally violated by asymmetric quantum strategies of minimal dimension. In contrast, one can also find examples of asymmetric Bell inequalities that are maximally violated by symmetric correlations. The implications of these findings on the geometry of the set of quantum correlations and the possibility of performing self-testing therefrom are briefly discussed.
12+7 pages, 3 figures, 6 tables, and several proofs. Comments are welcome!
arXiv:2601.02893 [quant-ph]
(or arXiv:2601.02893v3 [quant-ph] for this version)
2026 Quantum Sci. Technol. 11 035003
https://doi.org/10.1088/2058-9565/ae6bb1 DOI(s) linking to related resources
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