Bell's inequality

Einstein's objection

Entangled qubits always agree when measured the same way, no matter how far apart. Einstein saw nothing spooky there: perhaps each pair simply left the source carrying matching instruction lists — hidden variables — like two envelopes sealed with the same letter. Randomness would just be our ignorance of the list.

For thirty years this seemed like philosophy: both stories predicted identical experiments. Then in 1964 John Bell found a scenario where they disagree — measurably.

Three directions, one inequality

Give Alice and Bob each one qubit of a Bell pair, and let each independently measure in one of three directions, 120° apart. If hidden instruction lists exist, each qubit carries three pre-decided answers (one per direction). With only two possible values for three answers, at least two of the three must match — and a short counting argument then forces the pair to agree at least 1/3 of the time when directions are chosen at random.

Quantum mechanics predicts agreement of only cos²(60°) = 1/4 when measuring 120° apart. One quarter is less than one third — the two theories finally part ways at a number you can go out and measure.

The verdict is in: decades of increasingly airtight experiments (honored with the 2022 Nobel Prize in Physics) find 1/4. Nature really does not decide outcomes in advance; the correlations of entanglement have no classical explanation.

See it in the simulator

The Bell's Inequality preset builds the Bell pair and then tilts qubit 1's measurement basis by 120° using an Ry gate — rotating the basis before a standard measurement is exactly what 'measuring in a different direction' means. Run it and add P(00) + P(11): you'll get 0.25 on the nose. Try Ry angles of 0 and π to see perfect agreement and perfect anticorrelation at the extremes.

One caution from the entanglement lesson still applies: neither side can use this to signal. Alice's outcomes alone are a fair coin regardless of what Bob does. The magic is only visible when the two sides later compare notes.