bibkey: baumer2024trianglelocal
authors: Elisa Bäumer; Victor Gitton; Tamás Kriváchy; Nicolas Gisin; Renato Renner
year: 2024
title: "Exploring the local landscape in the triangle network"
doi: 10.48550/arXiv.2405.08939
url: https://arxiv.org/abs/2405.08939v1
claim: "Local models in the triangle network reach a fully symmetric four-outcome distribution with p(A=B=C) = 1/4; the paper conjectures that local fully symmetric distributions have maximal p(A=B=C) very close to 1/4, and asks as an open problem whether a local fully symmetric distribution with p(A=B=C) > 1/4 exists. From neural-network estimates it states that approximately s111(p) - 0.475 Delta_{l=1}(p) <= 0.289 (eq. ineq_l1) should hold for all local models."
strata_touched:
  - D5/S3/Quantum/Entanglement/TriangleSymmetricLocalRefutation
  - D5/S3/Quantum/Entanglement/TriangleInequalityL1Refutation
license: citation-only
triage: anchor

Exploring the local landscape in the triangle network

E. Bäumer, V. Gitton, T. Kriváchy, N. Gisin and R. Renner, arXiv:2405.08939 (v1 2024-05-14, the only version); Phys. Rev. A 111, 052453 (2025). Subject: quant-ph.

In the triangle network three independent sources α, β, γ each feed two of the parties Alice, Bob and Charlie. A distribution p(a, b, c) is local if

p(a,b,c) = ∫_{[0,1]^{×3}} dα dβ dγ p_A(a|β,γ) p_B(b|γ,α) p_C(c|α,β),

with α, β, γ uniform on [0, 1] and conditional distributions p_A, p_B, p_C (eq. trilocal). A distribution with four outcomes per party is fully symmetric if it is invariant under permutations of the parties and under joint permutations of the outcomes; s₁₁₁ = p(A = B = C). The paper constructs local fully symmetric models with s₁₁₁ = 1/4, estimates local bounds with neural networks, and states in its conclusion:

Open problem. Does there exist a distribution p(A,B,C) that is local in the triangle network and fully symmetric such that p(A=B=C) > 1/4?

To build Bell-type inequalities it penalises asymmetry by Δ_l = Σ_X Σ_{(a,b,c) ∈ I_X} |M_X − p(a,b,c)|^l, with I_X the outcome triples of type 111, 112 and 123 (4, 36 and 24 of them) and M_X the mean of p over I_X, and states that approximately

s₁₁₁(p) − 0.475 Δ_{l=1}(p) ≤ 0.289, s₁₁₁(p) − 5.211 Δ_{l=2}(p) ≤ 0.316,

should both hold for all local models (eqs. ineq_l1, ineq_l2).

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2405.08939
• URL: https://arxiv.org/abs/2405.08939v1 (source of v1 retrieved 2026-09-29).
• Location: §2 (“Setup: locality and symmetry”) for eq. trilocal and the definition of fully symmetric distributions; the section on inequalities for eq:ineq_l1 and eq:ineq_l2; the Conclusion for the open problem.