Bell's theorem

John Stewart Bell was born in Belfast in 1928 to a working-class family. He left school at sixteen because there was no money for more, and took a job as a laboratory technician at Queen's University — where the staff, recognising what they had, let him attend lectures. He eventually took a degree in experimental physics, then a second in mathematical physics, then a doctorate, and ended up at CERN designing particle accelerators. That was his job. Foundations of quantum mechanics was his hobby — his own word — pursued in evenings and on sabbaticals, and kept deliberately separate from the work he was paid for. Part of the reason for that separation was professional. In the 1950s and 60s, worrying about the foundations of quantum mechanics was regarded as slightly disreputable: a sign of philosophical confusion, or of not having enough real work to do. Bell was careful.

John Bell at a blackboard, gesturing towards written mathematics. — John Bell at CERN, where his day job was accelerator physics; the theorem was done in his own time. He spent much of his life arguing that the foundations of quantum mechanics deserved serious attention, at a period when saying so was a good way to be thought unserious.

There was a specific reason the question was considered closed. In 1932 John von Neumann published a proof that hidden-variable theories are impossible. Von Neumann was among the most formidable mathematicians of the century; the proof appeared in a rigorous and intimidating book; and for three decades it was taken to have settled the matter. If hidden variables are mathematically impossible, then Einstein's programme is not merely unfashionable but incoherent, and there is nothing left to argue about. Max Born said the proof showed that no concealed parameters could be introduced. It was cited, repeatedly, as the end of the discussion.

You might think

Von Neumann proved that hidden-variable theories are impossible.

Actually

He proved something much weaker than was claimed for it. The mathematics is correct; the trouble is an assumption smuggled into the setup — that a hidden-variable theory must assign values to sums of observables additively, even for observables that cannot be measured together. No sensible hidden-variable theory has that property, and Bohm’s does not. Bell put it bluntly: von Neumann’s proof was not merely false but foolish. And here is the uncomfortable part — Bell was not first. Grete Hermann, a mathematician and philosopher who had studied under Emmy Noether, published exactly this criticism in 1935, three years after the proof appeared. She was ignored for over forty years, and her paper was not widely known until the 1970s. A generation of physicists believed a question was closed because nobody read the person who had reopened it.

The bound is not a quantum quantity — no local theory of any kind can pass it.

Loading the detectors…

The CHSH form of the same result, which is what experiments actually measure. The local realist maximum is 2, and the local hidden-variable model in this simulation reaches exactly that. Quantum mechanics predicts 2√2 ≈ 2.828 — the Tsirelson bound, which is the most any quantum state can achieve.

Real experiments use a different but equivalent form, due to Clauser, Horne, Shimony and Holt in 1969 — the CHSH inequality. It is preferred for two practical reasons. It needs only two settings per side rather than three, which halves the apparatus. And it is stated in terms of correlations rather than counts, so it tolerates detectors that miss most of the particles — which every real detector does. It combines four correlation measurements into one number: S = |E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)| Every local realist theory gives S ≤ 2. Quantum mechanics predicts 2√2 ≈ 2.828 at the optimal settings. And 2√2 is not merely the quantum answer — it is the maximum any quantum state can reach, a limit known as the Tsirelson bound. Which raises a question nobody has fully answered: why does quantum mechanics violate the classical bound, but not maximally? Theories exist that would give S = 4 without permitting signalling. Nature apparently declines to use them.

The three assumptions

Everything the theorem rules out depends on exactly what it assumes, and it is worth naming all three — because the experiments in the next lesson are organised around closing them one at a time. Realism. Measurement outcomes are determined by properties the system already possesses. The particle has an answer before it is asked. Locality. What happens at one detector does not influence the other. Made rigorous by separating the wings and choosing settings so late that no signal at light speed could get across in time. Freedom of choice. The detector settings are not themselves determined by whatever produced the particles. If some common cause in the past fixed both the settings and the hidden variables, the derivation fails. Violating the inequality means at least one of these is false. The theorem does not tell you which — and that ambiguity is not a weakness in the argument, it is the actual situation. Every interpretation of quantum mechanics is, among other things, a choice about which one to abandon.

You might think

Bell proved that hidden variables are impossible, and that quantum mechanics is non-local.

Actually

Neither. Hidden variables are alive. Bohmian mechanics is a fully worked hidden-variable theory that reproduces every quantum prediction and survives Bell comfortably — by being frankly, explicitly non-local. What Bell rules out is the conjunction of hidden variables and locality. And the theorem does not single out locality as the casualty. It shows that at least one of realism, locality and free choice must go; which one is an interpretive choice. Many-worlds, for instance, claims to preserve locality by abandoning single definite outcomes. Most importantly, the theorem is not about quantum mechanics at all. It is a constraint on any local realist theory whatsoever. If quantum mechanics were overturned tomorrow, Bell’s inequality would stand unchanged, and the measured violations would still demand an explanation.

The experiment

The paper nobody read

John Stewart Bell · 1964, published November 1964 · Written on sabbatical at SLAC and Brandeis, away from his accelerator work at CERN

The question: Can any theory in which particles carry definite properties, with no influence between distant measurements, reproduce the correlations quantum mechanics predicts for entangled pairs?
The apparatus: None — a two-page derivation. The essential move was to consider correlations at unmatched detector settings, which nobody had examined because both theories agree at the aligned, perpendicular and opposed configurations that everyone naturally tried.

Theory predicted

Bell was a realist by inclination, admired Bohm’s theory, and thought Einstein had been treated poorly by the historical record. He expected to find that local hidden variables could reproduce the correlations, or failing that, that experiments would come out on Einstein’s side.

They measured

Local hidden variables cannot reproduce them. Any such theory obeys an inequality that quantum mechanics violates by a factor approaching 1.87 in the counting form, and reaches 2√2 against a bound of 2 in the CHSH form.

How sure could they be? It is a proof, so exact — but its significance is that it converts a philosophical dispute into a measurable quantity. The paper appeared in Physics Physique Fizika, a new journal that paid its contributors and folded after four issues; Bell never claimed his fee. It went essentially uncited for five years.

Why it mattered

Thirty years of argument about whether quantum mechanics is complete turned out to have an experimental answer, and the answer was obtainable with equipment that already existed. Bell built the instrument that could refute his own preference, published it, and urged people to use it. Nobody did for eight years.

For me, it is so reasonable to assume that the photons in those experiments carry with them programs, which have been correlated in advance, telling them how to behave. This is so rational that I think that when Einstein saw that, and the others refused to see it, he was the rational man.
John Bellinterview with Jeremy Bernstein, 1988 — after the experiments had gone against him

Grete Hermann

Grete Hermann took her doctorate under Emmy Noether at Göttingen in 1925, then moved into philosophy of physics. In 1935 she published a detailed analysis of von Neumann’s impossibility proof and identified precisely the flawed assumption Bell would identify thirty years later — that the theory must be additive over observables that cannot be jointly measured. Her paper appeared in a philosophy journal, was not read by physicists, and vanished. She spent the later 1930s in the German resistance, escaped to Denmark and then Britain, and after the war worked mainly in education and politics. Bell learned of her work only after publishing his own. It is a clean illustration of a general problem: a correct result in the wrong journal, by someone outside the field, addressed to an audience that was not listening.

1932Von Neumann publishes his impossibility proof. The question is considered closed.
1935EPR. And Grete Hermann finds the flaw in von Neumann, to no effect whatever.
1952Bohm builds a working hidden-variable theory — which should have ended the impossibility claim, and did not.
1964Bell’s theorem, in a journal that folds after four issues.
1966Bell publishes his critique of von Neumann — written earlier, delayed two years by an editorial mishap.
1969CHSH give the form experiments can actually use.
1972Freedman and Clauser perform the first test.

Where this leaves us

In 1964 this is still a calculation. Somebody has to build the thing. That means producing entangled pairs reliably, sending them far apart, setting detectors at the awkward intermediate angles nobody had tried, and counting coincidences for long enough to beat the statistics. It took eight years for a first attempt, another decade to do it convincingly, and another thirty years to do it without leaving a determined sceptic anywhere to stand. Because a local realist can always ask a further question. Were the detector settings chosen late enough that no light-speed signal could have carried them across? Did the detectors catch a representative sample, or only a convenient subset? Were the settings really free, or could some common influence have fixed both them and the particles? Each of those is a loophole, each is legitimate, and closing them turns out to be the interesting part of the story — because it forced experimenters to do things like slaving detector settings to the light of distant quasars, so that any conspiracy would have had to be arranged before the Earth existed.