Bell / CHSH · entanglement breaks local realism simulation

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?

▶ Run the simulationSee the measured result

Units: dimensionless (CHSH correlator S = 2√2, Tsirelson's bound)

How the lab tests it

Emit 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.

What it checks

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