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

TL;DR

This work examines whether party-permutation symmetry (PPI) in Bell inequalities can be preserved without increasing the local Hilbert-space dimension of the maximizing quantum strategy. It demonstrates that symmetric strategies in minimal dimension suffice for the CHSH and CGLMP families, but identifies concrete symmetric inequalities in low settings where maximal violations require asymmetric or higher-dimensional strategies, revealing a symmetry-dimension trade-off. The authors also show that asymmetric strategies can yield symmetric correlations and discuss the geometric consequences for the quantum set, including flat regions that obstruct self-testing. Overall, the results clarify when symmetry is a free resource and when asymmetry becomes essential for maximal Bell violations, with implications for device-independent certification and the structure of quantum correlations.

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.

Figures (3)

• Figure 1: Schematic showing different pathways to obtain a purified, symmetric quantum strategy (PSQS) $\tilde{\mathfrak{Q}}$ from any quantum strategy (QS) $\mathfrak{Q}$ producing a symmetric correlation $\vec{P}_{\hbox{$\leftrightarrow$}}$. In general, this involves performing step 1. Naimark dilation + purification (horizontal solid arrow), and 2. symmetrization (vertical solid arrow) in either order. In both cases, the local HSD is at least doubled. However, if the initial strategy $\mathfrak{Q}$ consists of a pure state and PVMs, it may even be possible to obtain a PSQS via a local unitary transformation (dashed arrow), which preserves the local HSD: an example being the transformation of the strategy of \ref{['eq:max_CHSH_strategy']} to that of \ref{['eq:CHSH ss strategy']}.
• Figure 2: Maximal Bell value of $I_S(\alpha)$ for $\alpha\in[1.5,3]$ under various constraints. From bottom to top, we have, respectively, the local bound of $2\alpha+5$ (red, dashed), the symmetric qubit upper bound (blue, dotted) computed using the method described in \ref{['App:Techniques']}, and the general quantum bound (yellow, solid), which is attainable using a two-qubit QS involving a degenerate observable for one of the parties.
• Figure 3: Measurement directions $\vec{a}_k$ (solid line) and $\vec{b}_k$ (dashed line) for $k=0,1,2,3$ corresponding to the QS described in \ref{['eq:J42_strategy']}. Notice that each $\vec{b}_k$ may be obtained from the corresponding $\vec{a}_k$ by performing a mirror reflection about the $x-z$ plane (the pale blue plane), making it evident that the QS is asymmetric. Surprisingly, the resulting correlation is symmetric and gives a Bell value $\approx 0.6012$ for the $J^{42}_{4422}$ inequality, higher than the 0.5682 bound achievable by any qubit SQS.

Theorems & Definitions (20)

• Definition 1: Symmetric correlation
• Definition 2: Symmetric Bell inequality
• Definition 3: Symmetric quantum strategy
• Proposition 1
• Proposition 2
• Theorem 3
• Lemma 4
• proof
• proof
• Proposition 5