Can the correlations between two entangled particles be explained by any theory in which each particle carries predetermined answers fixed at emission (a 'local hidden variable') — or does nature refuse to be locally real?
▶ Launch the interactive simulationEmit two spin-½ particles in the singlet Bell state |ψ⁻⟩ = (|↑↓⟩ − |↓↑⟩)/√2 toward two distant analyzers. Alice measures spin along an axis at angle a, Bob along b, each getting ±1; quantum mechanics gives the correlation E(a,b) = ⟨AB⟩ = −cos(a−b). Monte-Carlo-sample the ±1 outcomes from the QM joint distribution (never the formula for S), estimate the four correlators from the counts, and form the CHSH combination S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′) at the optimal angles a=0°, a′=90°, b=45°, b′=135°, swept across 24 seeds. Controls: an explicit local-hidden-variable model (shared emission angle λ, deterministic local response), and a Werner state of tunable visibility V.
Tsirelson's bound S = 2√2 ≈ 2.828 — recovered from the sampled counts to ±0.01, with EVERY seed exceeding the local-realist bound |S| ≤ 2 by ≈ 0.83 (the violation is unambiguous on each single run). This is Bell's theorem, measured: no theory of local hidden variables can reproduce quantum mechanics. Two controls falsify local realism — the explicit local-hidden-variable model, scanned over all analyzer settings, never climbs past S = 2 (it saturates there but cannot reach 2√2); and a Werner state gives S(V) = 2√2·V, so the violation switches off CONTINUOUSLY and crosses the classical bound exactly at V = 1/√2 ≈ 0.707. The bound 2√2 is never plugged in — it emerges from the sampled outcomes. Bell (1964) / CHSH (1969) / Tsirelson (1980), confirmed by Aspect (1982) and the loophole-free tests (2015), Nobel Prize 2022 — the lab's first world about quantum entanglement and nonlocality