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Bell-locality threshold of two-qubit Werner states

For ρv=v∣ψ−⟩⟨ψ−∣+(1−v)I/4\rho_v=v|\psi^-\rangle\langle\psi^-|+(1-v)I/4, determine the exact critical visibility for Bell locality under all single-copy local projective measurements; formulate arbitrary POVMs as a separate threshold.

Source-reported statusAs of 2026-09-04Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

For ρv=v∣ψ−⟩⟨ψ−∣+(1−v)I/4\rho_v=v|\psi^-\rangle\langle\psi^-|+(1-v)I/4, determine the exact critical visibility for Bell locality under all single-copy local projective measurements; formulate arbitrary POVMs as a separate threshold.

Lean 4

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Background

A two-qubit Werner state mixes a singlet with white noise. Entanglement and Bell nonlocality are different resources: the state is entangled for v>1/3v>1/3, but some entangled states still admit local hidden-variable models.

Acín–Gisin–Toner proved that the projective-measurement locality threshold is vproj=1/KG(3)v_{\rm proj}=1/K_G(3), with KG(3)K_G(3) the order-three Grothendieck constant. Thus exact determination is a precise mathematical problem, not just a search for another Bell inequality. The threshold for all POVMs must be kept separate.

Published progress (February 2026): Designolle–Vértesi–Pokutta improve certified bounds for finite-order Grothendieck constants and clearly distinguish exact certificates from heuristic candidates. The exact order-three value remains unknown. No numerical interval is displayed here without a separately maintained certificate ledger.

The operational setting is one copy, ordinary classical-input Bell tests and no postselection or auxiliary entanglement. Filtering, collective measurements on several copies and network activation change the question; none should be silently included in this threshold.

Progress & source assessment

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

published: New certified finite-order Grothendieck bounds; exact order-three value remains unknown, and heuristic candidates are distinguished from certificates. Evidence

References
  1. [1]Acín, Gisin & Toner, 'Grothendieck's constant and local models for noisy entangled quantum states', PRA 73, 062105 (2006) link
  2. [2]Designolle, Vértesi & Pokutta, 'Better bounds on finite-order Grothendieck constants', PRA 113, 022401 (2026) link

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