Entangled spin pairs leave the source. Alice and Bob each pick one of two analyzer angles at random and record +1 or −1. Switch between quantum physics and a local hidden-variable model and see which one breaks the |S| ≤ 2 limit.
Changing an angle resets the counts. The Bell angles (0°, 90°, 45°, 135°) give the largest possible violation.
| Setting | Pairs | E measured | E theory |
|---|
S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′). Any local theory gives |S| ≤ 2. Quantum mechanics reaches 2√2 ≈ 2.83.
━ quantum: −cos Δθ ┅ local: linear ● measured
P(A = +1) stays at 50% whatever Bob chooses. On her own, Alice sees only noise; the correlation shows up only when the two notebooks are compared.
When they measure spins along the same direction, Alice and Bob always get opposite results, no matter how far apart they are. For Einstein, either the results were already written into the particles at the source, or one measurement affected the other at a distance: "spooky action at a distance".
The "local hidden variables" mode is exactly the first option: each pair carries a secret angle λ, and each side answers using only λ and its own analyzer angle. It mimics the perfect correlations, but never gets past |S| = 2. Nature does, as shown by Aspect (1982) and the loophole-free tests of 2015.
Each side, on its own, sees +1 and −1 half the time, regardless of what the other did. The correlations only appear when the records are compared over an ordinary channel, which is limited by the speed of light. Nothing useful travels faster than light.