What this page shows. This is the experiment that proved nature is not local: a measurement on one atom here changes what its twin does over there, with nothing passing between them. The setup is Bell's. An oven sends out pairs of atoms in the singlet state. A magnet on each side measures its atom's spin along an axis you choose. The ledger counts how often the two answers agree. The switch at the top decides the pairs under one of two suppositions, and only one of them can match what the laboratory measures.
The paradox of Einstein, Podolsky and Rosen. In 1935 EPR asked whether quantum mechanics is a complete description of reality, and pointed at one plain fact about these pairs: set both magnets the same way and the two answers are always opposite. Measure A and you know B's answer for certain, without touching B. By EPR's rule that makes B's spin an element of physical reality: it was settled before anyone looked. But quantum mechanics says neither spin has a value until it is measured. So how does B know what A found? Einstein would not accept "spooky action at a distance". The sensible conclusion, EPR said, is that the wave function is not the whole story: each atom carries something from the oven that the theory leaves out. That something is called a hidden variable. For thirty years this was treated as a matter of taste. In 1964 John Bell showed it was a matter of fact, and one that could be measured.
Local hidden variables, stated plainly. Call the hidden variable λ. Each pair leaves the oven carrying its λ. Magnet A's answer is fixed by its own setting and λ, nothing else; Magnet B's answer by its own setting and λ, nothing else. That is Bell's requirement of locality: A's answer does not depend on how B is set, nor B's on A. And since aligned magnets always disagree, B carries the opposite answer to every question A could be asked. It does not matter what λ is, how many numbers it holds, or how the oven picks it. Bell's inequality holds for every such theory.
The test: aligned, then 60° each way. Start with both magnets at 0°. The answers are always opposite: 0% agree. Turn Magnet A alone to 60°. Now some pairs agree. B at 0° still gives the opposite of what A would give at 0°, so these are exactly the pairs whose λ makes A answer differently at 60° than at 0°. Call that share a. Put A back to 0° and turn Magnet B alone to −60° (300° on its ring). The pairs that agree now are the ones whose λ makes B answer differently at −60° than at 0°. Call that share b. Now set both: A at 60°, B at −60°, 120° apart. For a pair to agree, at least one of its two answers must have changed from the aligned case. And locality says whether A's answer changed was fixed by λ, not by where B is set. So the share that agree at 120° is at most a + b. Two turns that each change a quarter of the answers can change at most half of them. That is Bell's inequality. It needs no physics, only counting.
What the laboratory finds. Quantum mechanics predicts, and every experiment confirms: at 60° apart the pairs agree 25% of the time; at 120° apart, 75%. Twenty-five and twenty-five cannot make seventy-five. No set of hidden variables fixed at the oven can give those three numbers at once, as long as each answer depends only on its own magnet. In Bell's words: no local hidden-variable theory can reproduce all the predictions of quantum mechanics. The experiment has been done with photons, ions and atoms, by Clauser, Aspect, Zeilinger and others, with the settings chosen while the pairs were in flight and the detectors too far apart for any signal to cross in time. The answer is 75%.
Why no hidden variables. Suppose the answers were fixed at the oven. Then each atom left carrying a list: up or down at 0°, up or down at 60°, up or down at −60°, and so on for every axis, and its twin carried the opposite list. A list is just a list. Turning Magnet A to 60° reads a different line of A's list, and in a quarter of the pairs that line differs from the 0° line. Turning B to −60° reads a different line of B's list, and in a quarter of the pairs that one differs too. For the two atoms to agree at 120°, at least one list must differ between its 0° line and its turned line, which can happen in at most half the pairs. Nature says three quarters. So the lists do not exist. There was no answer written down for the 60° question before the 60° question was asked.
Why not local. Now suppose instead that the answers are made on the spot, when each atom reaches its magnet. Then the only way to get three quarters at 120° is for the two answers to be made together: the share that agree when both magnets are turned is more than the two single turns can account for, so turning B must have changed what happened at A, or the other way round. But A and B are far apart, and the settings can be chosen after the atoms have left the oven, too late for any signal to cross from one magnet to the other before the atoms arrive. Whatever ties the two answers together is not travelling between them. That is what nonlocal means: not that something moves faster than light, but that the pair does not behave as two separate things with separate facts.
The same point, once more. Bell's inequality assumes two things at once: that the answers are already settled (hidden variables), and that each answer depends only on its own magnet (locality). Everything else about the hidden variables is left open: how many there are, what they are, how the oven picks them. The measured 25%, 25% and 75% break the inequality. So one of the two assumptions is false. Drop hidden variables and the answers are not real until measured, and yet two measurements made far apart still come out matched beyond what any shared past can explain. Drop locality and the atoms carry their answers, but one magnet's setting reaches across to change the other atom's answer. Either way the two atoms are not independent. Quantum mechanics takes the first road and calls the pair a single state. No theory can take neither.
Try it on the plates. Leave the switch on hidden variables. Here λ is a direction picked at the oven, and in one pair of every four a fixed answer: that atom answers the same way whatever the axis. Set both rings to 0° and let the plates fill: 0% agree. Set A to 60°: 25%. Set B to 300° as well, 120° apart: 50%, the two turns added and not a pair more. Now move the switch to quantum and do the same. At 60° apart, 25% again; at 120° apart, 75%. The ledger shows what each supposition expects beside what the plates recorded, and Fig. 4 adds a dot for every angle you try. Try 180° too: the hidden variables agree 75% of the time there, where real pairs always agree. That is not a flaw in this model. A local theory that matches nature at 60° cannot also match it at 180°, and one that matches at 180° can do no better than 33% at 60°. In the laboratory the dots fall on the solid curve everywhere.
What is left standing. Each answer on its own is still a coin toss, and neither plate can be used to send a message to the other, so nothing here outruns light. What falls is EPR's picture of two separate atoms, each carrying its own facts. The pair is one system until it is measured, however far apart its halves are. That is what nonlocal means. It is not a supposition any more; it is a measurement.