Trading symmetry for Hilbert space dimension in Bell-inequality violation

Hsin-Yu Hsu^1*, Gelo Noel M. Tabia^3,1,2, Mu-En Liu¹, Kai-Siang Chen¹, Tamás Vértesi⁴, Yeong-Cherng Liang^1,2

¹Department of Physics, National Cheng Kung University, Tainan, Taiwan
²Physics Division, National Center for Theoretical Sciences, Taipei, Taiwan
³Hon Hai (Foxconn) Research Institute, Taipei, Taiwan
⁴HUN-REN Institute for Nuclear Research, Debrecen, Hungary

* Presenter:Hsin-Yu Hsu, email:L28111566@gs.ncku.edu.tw

In quantum information, asymmetry, i.e., the lack of symmetry, is a resource that enables one to accomplish certain tasks that would otherwise be 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 of dentical quantum particles and explore when we can trade symmetry for a lower-dimensional quantum strategy in achieving the same Bell violation. For the family of symmetric Collins-Gisin-Linden-Massar-Popescu inequalities, we provide evidence showing that there is no such tradeoff. However, for several other Bell scenarios with a small number of dichotomic measurement settings, we have found examples where symmetric quantum strategies in the minimal Hilbert space dimension can only lead to a suboptimal Bell violation. 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.

Keywords: Quantum nonlocality, Quantum correlation, Symmetry, Dimension constraint