bibkey: hsuetal2026tradingsymmetry
authors: Hsin-Yu Hsu; Gelo Noel M. Tabia; Kai-Siang Chen; Mu-En Liu; Tamás Vértesi; Nicolas Brunner; Yeong-Cherng Liang
year: 2026
title: "Trading symmetry for Hilbert-space dimension in Bell-inequality violation"
doi: 10.1088/2058-9565/ae6bb1
url: https://arxiv.org/abs/2601.02893v4
claim: "The paper asks whether symmetric quantum strategies, with identical measurements on a swap-invariant state, reach the maximal quantum violation of party-permutation-invariant Bell inequalities in the minimal Hilbert-space dimension; it gives a family of identical Fourier-type measurements for the CGLMP inequality, finds Bell inequalities where symmetric strategies in minimal dimension are suboptimal, and conjectures that its symmetric form of the two-setting SATWAP inequalities reaches the maximal quantum value 2(d-1) with these measurements and a symmetric maximally entangled state for every d."
strata_touched:
  - D5/S3/Quantum/Contextuality/SymmetricSatwapMaximalViolation
license: citation-only
triage: anchor

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

H.-Y. Hsu, G. N. M. Tabia, K.-S. Chen, M.-E. Liu, T. Vértesi, N. Brunner and Y.-C. Liang, arXiv:2601.02893v4 (17 June 2026, the published version); Quantum Science and Technology 11(3) (2026), 035003. Quotations are from the v4 TeX source, with the macros \ketbra and \intv{d} (that is, $[d]$) written out and citations omitted.

Symmetric quantum strategy (Definition def:Symmetric strategy): both parties perform the same measurements, $M_{a|x}^B=M_{a|x}^A$, on a shared state with ${\rho }_{BA}={\rho }_{AB}$, where ${\rho }_{BA}:=S{\rho }_{AB}{S}^{†}$ and $S:={\sum }_{ij}|i⟩⟨j{|}_A\otimes |j⟩⟨i{|}_B$.

The measurements:

$M_{a|0}^A=|a⟩⟨a|,M_{a|1}^A=U|a⟩⟨a|U^{†},U:=TW,$ … $T:=-1⨁{\mathbb{I}}_{d-1}$ and $W:={\sum }_{i,j\in [d]}\frac{|i⟩⟨j|}{d\sin [(i-j-\frac{1}{2})\frac{\pi }{d}]}$.

The correlators: $⟨A_x^k{B}_y^{\ell }⟩={\sum }_{a,b=0}^{d-1}{\omega }^{ak+b\ell }P(a,b|x,y)$ with $\omega ={\mathrm{e}}^{2\pi i/d}$.

The symmetric SATWAP expression:

${\tilde{\mathcal{I}}}_d={\sum }_{k=1}^{d-1}\left[{\omega }^{-2k}\right(a_k⟨A_1^{-k}{B}_1^{-k}⟩+a_k^{\ast }⟨A_1^{-k}{B}_2^{-k}⟩+a_k^{\ast }⟨A_2^{-k}{B}_1^{-k}⟩)+{\omega }^{-3k}{a}_k⟨A_2^{-k}{B}_2^{-k}⟩]$, where $a_k=\frac{(1-i)}{2}{\omega }^{\frac{k}{4}}$.

The conjecture:

Our computation results show that for the symmetric inequalities of Eq. (symSATWAP), the measurement of Eq. (CGLMP-Opt), in conjunction with a symmetric maximally entangled two-qudit state, indeed reproduces the maximal quantum violation of $2(d-1)$ for $d\le 100$. Together with the fact that this maximal quantum violation self-tests a maximally entangled two-qudit state, we know that the symmetric family of SATWAP inequalities, Eq. (symSATWAP), also cannot exhibit a trade-off for $d\le 100$. Moreover, we conjecture that this holds true for any higher $d$ as well.

The measurements are labelled by settings $0,1$ and the SATWAP expression by settings $1,2$. For $d=2$ both assignments of the settings reach the value $2$; for $d=3,4,5,8$ the largest eigenvalue of the Bell operator is $2(d-1)$ only when setting $0$ is assigned to setting $1$ (floating-point computation).

Verified locator

Source: https://arxiv.org/abs/2601.02893v4

• URL: https://arxiv.org/abs/2601.02893v4 (the e-print is a tar archive; member SymmetricBIV-arXiv_v2.tex, 2736 lines, md5 6116dc142244f252c16a835d291626cf):
  • Definition def:Symmetric strategy, l. 285–297;
  • the measurements, Eq. Eq:CGLMP-Opt, l. 488–498;
  • the correlators, Eq. Eq:Correlators:d, l. 513–519;
  • the symmetric SATWAP expression, Eq. Eq:symSATWAP, l. 523–530;
  • the conjecture, l. 532.
• DOI: 10.1088/2058-9565/ae6bb1 (https://doi.org/10.1088/2058-9565/ae6bb1; Crossref lists Quantum Science and Technology, volume 11, issue 3, article 035003, published 5 June 2026).