Exact critical visibility of two-qubit Werner states

Determine the exact critical visibility v_c for which the two-qubit Werner states ρ_v = v |ψ⟩⟨ψ| + (1 − v) I/4, with |ψ⟩ = (|↑↓⟩ − |↓↑⟩)/√2, cease to admit a local hidden-variable model under all projective measurements; that is, precisely identify the threshold v_c at which these states transition from Bell-local to non-local.

Background

Werner states are mixtures of the Bell singlet and white noise that illustrate the inequivalence between entanglement and non-locality: some entangled mixed states remain Bell-local. For two qubits, these states are separable for visibility v ≤ 1/3 and entangled otherwise. Werner constructed a local hidden-variable model for projective measurements at v = 1/2, establishing locality in that regime. Despite extensive work providing lower and upper bounds on the transition to non-locality, the exact threshold visibility v_c at which two-qubit Werner states become non-local under all projective measurements has not been determined.

The present paper develops a general machine-learning-based method to construct local hidden-variable models for arbitrary multipartite states and continuous measurements and reports a numerical estimate for v_c. However, the exact analytic value remains unresolved, motivating a precise determination of v_c.