Bell's Theorem
Bell's theorem is the result, proved by John Stewart Bell in 1964, that no theory in which measurement outcomes are determined by local variables carried by the particles can reproduce all the statistical predictions of quantum mechanics. It converts what had been a philosophical dispute — the Einstein–Podolsky–Rosen argument of 1935 — into a question decidable by experiment.
What the theorem actually says
Bell considered a pair of particles prepared together and then measured apart, each measurement made with a setting the experimenter chooses. He assumed:
- A hidden variable λ carried by the pair, which together with the local setting fixes the local outcome or its probability;
- Local causality: the outcome at one wing does not depend on the setting chosen at the other;
- Measurement independence (also called statistical independence or "free choice"): the distribution of λ does not depend on which settings are chosen.
From these he derived an inequality that the correlations between the two wings must satisfy. The most-used form is the CHSH inequality of Clauser, Horne, Shimony and Holt (1969): a particular combination S of four correlation measurements must satisfy |S| ≤ 2 for any theory of the above kind. Quantum mechanics predicts values up to 2√2 ≈ 2.828 (Tsirelson's bound) for suitably chosen settings, and it is this excess that experiment tests.
Two things the theorem does not say deserve emphasis, because both are commonly misreported:
- It does not show that quantum mechanics permits signalling faster than light. The no-signalling theorem holds: the marginal statistics at either wing are independent of the distant setting, and the correlations only become visible when the two records are brought together classically.
- It does not by itself show that "hidden variables are impossible". Bohmian mechanics is an explicit hidden-variable theory that reproduces quantum mechanics exactly; it does so by being nonlocal, which is what Bell's theorem requires of it.
The experiments
Freedman and Clauser reported the first violation in 1972. Alain Aspect's Orsay experiments of 1981–82, including the 1982 version with acousto-optic switches changing the analyser settings while the light was in flight, gave violations of many standard deviations.
These early tests left loopholes: the detection (or fair-sampling) loophole, since only a small fraction of emitted pairs were detected and the sample might be unrepresentative; and the locality loophole, if the settings and outcomes at the two wings were not spacelike separated. Each was closed individually over the following decades, but closing both at once required detectors efficient enough and separations large enough that it was not achieved until 2015, when three groups did so independently: Hensen and colleagues at Delft, using entangled nitrogen-vacancy centres 1.3 km apart with event-ready entanglement swapping; Giustina and colleagues in Vienna; and Shalm and colleagues at NIST, both using high-efficiency photon detectors. All three reported violations.
The remaining assumption is measurement independence. The 2017 "Cosmic Bell" experiments used the colours of photons from distant stars, and later from high-redshift quasars, to set the analysers, pushing any conspiracy between settings and source back billions of years; the 2018 BIG Bell Test used choices made by about 100,000 human volunteers. Neither closes the loophole in principle, and neither can: rejecting measurement independence is always logically available. Clauser, Aspect and Anton Zeilinger shared the 2022 Nobel Prize in Physics for this programme.
What remains genuinely open
Local realism in Bell's sense is excluded by experiment. That much is now as secure as experimental physics gets, and any model proposed here that relies on the detection or locality loopholes is answering a question that has been closed.
What is not settled, and what honest accounts leave open:
- Which assumption to give up. Nonlocality (Bohm), no single outcomes (many-worlds), no pre-measurement values (Copenhagen and its descendants), retrocausality, and superdeterminism are all consistent with the data. Physicists disagree about which is least costly, and the disagreement is not empirical at present.
- Superdeterminism — denying measurement independence — is not refuted, and its serious advocates, notably Gerard 't Hooft and Sabine Hossenfelder, are not cranks. It is unpopular because it appears to make experimental science's basic assumption negotiable, not because it has been disproved.
- What the correlations mean physically remains the measurement problem, which is unsolved.
On this wiki
This wiki holds a substantial body of critical work on Bell tests. It is important to read it with the chronology in mind: most of it predates 2015, and much of it turns on loopholes that the loophole-free experiments closed.
- Caroline H Thompson was the best-known critic of the fair-sampling assumption; her "What Really Happens in Those Bell Correlation Experiments?" and "Timing, 'Accidentals' and Other Artifacts in EPR Experiments" argue that subtraction of accidentals and coincidence-window choices manufacture part of the apparent violation. The detection-loophole part of this argument was answered by the 2013–2015 high-efficiency experiments; the methodological point about coincidence windows retains some force as a caution about older data.
- Al F Kracklauer pursues a local-realistic reconstruction in "A Local-Real Model of EPR Correlations" and "Nonlocality, Unreality, and Bell Theorem".
- Thomas Smid, "Classical Interpretation of EPR- Bell Test Photon Correlation Experiments".
- J F Geurdes, "A Counter-Example to Bell's Theorem with a 'Softened' Singularity" — a mathematical challenge to the derivation itself rather than to the experiments.
- Vesselin C Noninski, "Classical Bell's Inequalities"; Thomas E Phipps, "Measurement Theory via Hidden Variables Not Subject to Bell's Theorem"; Dan Brasoveanu, "Significant Facts Revealed by the EPR Paradox and Bell's Theorem".
- Susana F Huelga, "Optical Tests of Bell's Inequalities Closing the Poor Correlation Loophole", and Gary Felder, "Spooky Action at a Distance: An Explanation of Bell's Theorem", are expository or mainstream-aligned contributions in the same collection.
The general position taken across much of this material — that the correlations are real but the inference to nonlocality is not forced — is a defensible one, but the defensible version of it now has to run through superdeterminism, retrocausality or a denial of one of Bell's premises, not through detector inefficiency.