[challenge]: Sandwich-closing open Bell constants via state polynomials

Released by

Jie Wang (AMSS, Chinese Academy of Sciences) & Jin-Guo Liu (Hong Kong University of Science and Technology (Guangzhou))

Contact email

cacate0129@gmail.com

Method

Other

Challenge issue

Difficulty: ★ (rated by Jie Wang)

Background

The state-polynomial hierarchy (arXiv:2301.12513) settled two conjectured nonlinear Bell inequalities, and symmetry-adapted NPA proved the CGLMP maximal-violation conjecture for d = 3, 4 (arXiv:2112.10803). Many maximal-violation constants — tilted-CHSH families, nonlinear Bell functionals, uncertainty-relation constants (arXiv:2310.00612) — remain open, each a small theorem waiting for a certificate.

Research objective

A systematic sandwich-closing campaign: for each catalogued open constant, run symmetry-reduced state-polynomial relaxations downward (upper bounds) and variational see-saw strategies upward (lower bounds) until the gap closes to rational-certificate precision — both sides exact.

Verification plan

• Success gate (per constant): an exact rational SOHS certificate for the upper bound plus an explicit finite-dimensional strategy whose value matches to the certified precision — upper = lower is a self-validating sandwich that cannot be faked.
• Hope signal: gaps shrink monotonically with level but stall around 1e-6 — publish tight two-sided bounds; the stall pattern locates candidates for unattainment (cf. the non-closed quantum correlation set, arXiv:1703.08618).
• Pivot signal: every interesting constant stalls with a persistent gap — the campaign becomes evidence about which functionals exhibit non-attainment; pivot to studying that.

Why this may lead to research output

Each nontrivially closed constant is a standalone publishable theorem in quantum information, produced on an assembly line whose quality gate is exact arithmetic.

References

1. Klep, Magron, Volčič, Wang, State polynomials: positivity, optimization and nonlinear Bell inequalities, arXiv:2301.12513.
2. Ioannou, Rosset, Noncommutative polynomial optimization under symmetry, arXiv:2112.10803.
3. Uncertainty relations from state polynomial optimization, arXiv:2310.00612.
4. Slofstra, The set of quantum correlations is not closed, arXiv:1703.08618.

No files, tests, or entry points are identified in the issue. Start by locating the repository's state-polynomial, symmetry-reduction, and variational-optimization components, then review the cited papers and catalogue of open constants; done means closing a constant with an exact rational upper certificate and a matching finite-dimensional strategy.