Trading symmetry for Hilbert-space dimension in Bell-inequality violation

TL;DR

This paper investigates the relationship between symmetry and Hilbert space dimension in Bell inequality violations, revealing that certain symmetric inequalities require asymmetric strategies for maximal violation, impacting quantum correlation geometry.

Contribution

It demonstrates that some Bell inequalities necessitate asymmetric quantum strategies for maximal violation, challenging assumptions about symmetry's role in quantum correlations.

Findings

Symmetric Bell inequalities can only be maximally violated by asymmetric strategies.

Some Bell inequalities are maximally violated by symmetric correlations despite asymmetry in strategies.

No trade-off exists for the symmetric Collins-Gisin-Linden-Massar-Popescu inequalities.

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.