The Bell Inequality
Field Notes on Testing Locality
The Question
Is nature local? Do particles carry all the information they will ever need to produce measurement outcomes, like workers on an assembly line following instructions written before they left the factory? Or does measurement create reality, and does a measurement here instantly affect what is real there?
Bell's inequality is the test.
The Setup
Two particles in a singlet state. Separated. Alice measures one along axis a. Bob measures the other along axis b. They compare results. Over many runs, they compute the correlation function E(a,b).
If the world is local and real — if particles have definite values before measurement, if no influence travels faster than light — then:
|E(a,b) − E(a,c)| ≤ 1 + E(b,c)
This is the CHSH form. Bell's inequality. It is a mathematical certainty in any local hidden variable theory. Not a guess. Not a model. A theorem.
The Quantum Prediction
For a singlet state, quantum mechanics predicts:
E(a,b) = −a · b = −cos(θ)
where θ is the angle between measurement axes.
Choose a = 0°, b = 22.5°, c = 45°. Then:
E(a,b) = −cos(22.5°) ≈ −0.924 E(a,c) = −cos(45°) ≈ −0.707 E(b,c) = −cos(22.5°) ≈ −0.924
Left side: |−0.924 − (−0.707)| = 0.217 Right side: 1 + (−0.924) = 0.076
0.217 > 0.076.
The inequality is violated. By a wide margin. Quantum mechanics wins. Bell's theorem says no local hidden variable theory can reproduce this.
The Experiments
1972 — Clauser and Freedman: First experimental test. Violation confirmed. Error bars were large but the violation was clear.
1982 — Aspect, Grangier, and Roger: The definitive test. Switching measurement settings during the particles' flight — faster than light could travel between detectors. Still violated. Still no hidden variables.
2015 — Hensen et al.: The loophole-free Bell test. Both detection efficiency and locality satisfied simultaneously. No escape hatch.
2022 — Nobel Prize: Aspect, Clauser, Zeilinger. The question is settled.
What This Means
The universe is not locally real. One of the following must be true:
- Locality is false. Measurement outcomes are not independent at spacelike separation. Something connects them.
- Reality is false. Particles do not have definite properties before measurement.
- Both are false.
Most physicists choose option one, or some variant where "information" or "influence" is non-local but cannot be used for signaling. The no-communication theorem protects causality while non-locality runs free underneath.
Practical Takeaway
Bell's inequality is not an abstract curiosity. It is the resource certification protocol for quantum cryptography. If your encryption key violates Bell's inequality, it was generated by entangled particles and no eavesdropper could have intercepted it without breaking the entanglement.
Test the inequality. Verify the violation. Trust the universe.