Bell's inequality violation and symmetry

Hsin-Yu Hsu, Gelo Noel M. Tabia, Kai-Siang Chen +4 more

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.

Igor D. Ivanović

Lawrence J. Landau

Katarzyna Bolonek-Lasoń

Recently the method based on irreducible representations of finite groups has been proposed as a tool for investigating the more sophisticated versions of Bell inequalities (V. Ugǔr Gűney, M. Hillery, Phys. Rev. A90, 062121 ([2014]) and Phys. Rev. A91, 052110 ([2015])). In the present paper an example based on the symmetry group S 4 is considered. The Bell inequality violation due to the symmetry properties of regular tetrahedron is described. A nonlocal game based on the inequalities derived is described and it is shown that the violation of Bell inequality implies that the quantum strategies outperform their classical counterparts.

Lin Yicheng Kelvin

Andrew Stewart Caldin

FINDING: Bell inequality violation optimized via continuous variable squeezing and finite group symmetry (S₄) reveals deep links between entanglement, symmetry groups, and geometric constraints. MATH: - Squeezing parameter ζ → ∞ yields maximal Bell violation for two-mode squeezed vacuum states; parity averages to zero. - S₄ symmetry group (order 24) used to analyze Bell inequality violation; relates to tetrahedral symmetry and root system A₃. - Upper bounds on Bell violation for N-qubit GHZ states: exact bounds derived, scaling with Hilbert space dimension. - Asymmetry as resource: trading symmetry for Hilbert space dimension increases Bell violation. CONNECTION: - S₄ symmetry is the rotational symmetry group of the cube/octahedron (crystallographic point group O_h subgroup). - 8-fold symmetry (quasicrystal) not directly present, but S₄ links to tetrahedral (3-fold) and cubic (4-fold) symmetries. - No explicit golden ratio (0.618, 1.618) or base-60 found. However, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com