John Stewart Bell was born in Belfast in 1928 to a working class family. Bell became a theoretical physicist at CERN, the European particle physics laboratory near Geneva, where his day job was particle physics and the design of accelerators. In 1964, during a year's leave in the United States, he wrote a short paper on the EPR problem that some physicists have since ranked among the deepest discoveries in science.

The central idea

Bell's insight was that the EPR case, where both sides measure along the same direction, is too easy. Instruction cards explain it. The real test comes when the two sides measure along different directions. Bell showed that any local hidden variable theory puts a strict limit on how strongly results can be correlated across different pairs of settings. That limit is expressed as an inequality, a statement that some combination of agreement rates can be no larger than a certain number. Quantum mechanics predicts that entangled particles break this limit for certain choices of settings. Both cannot be right, and experiment can decide.

The result is called Bell's theorem. It is not a claim about what nature does but a proof about two kinds of theory. Bell's theorem says no local hidden variable theory can reproduce all the predictions of quantum mechanics. Whether nature follows quantum mechanics is a separate, experimental question.

A game with instruction cards

The physicist N. David Mermin, at Cornell, found a version of Bell's argument in 1981 that needs only counting.

A source sends particles in pairs to two distant stations. Each station has a dial with three settings, labeled 1, 2 and 3, and a lamp that flashes either red or green when a particle arrives. For each pair, the operators at the two stations choose a setting independently and at random, and record the color. Over many runs two facts emerge. First, whenever both stations happen to choose the same setting, the lamps always flash the same color. Second, over all runs, with settings chosen at random, the lamps agree only half the time.

Can instruction cards explain both facts? To guarantee matching colors on matching settings, each pair must carry identical cards, each listing a color for every setting, such as red, red, green. Now count. With random choices there are nine equally likely setting combinations. For the card red, red, green, the lamps agree when both choose 1, both choose 2, both choose 3, or when one chooses 1 and the other 2, in either order. That is five of the nine. Try any card with two of one color and one of the other: by the same count, five of nine. A card with all three the same gives agreement every time. So every possible instruction card gives agreement at least five times in nine, which is more than half.

No mixture of cards can bring the average below five ninths. Yet quantum mechanics, for entangled photons whose filters are set at 0, 60 and 120 degrees, predicts agreement exactly half the time. Because cards stand for any local hidden variables, nothing local and realistic can match it.

Why the argument is so strong

The card argument assumed almost nothing about how particles work. The bound comes from counting alone, so it applies to any theory in which results are fixed locally in advance. That generality is why Bell's theorem matters. If experiment shows agreement near one half, then at least one assumption in the card picture must fail: locality, realism in the narrow sense, or measurement independence.

From theorem to test

Bell's original inequality assumed perfect correlations that no real apparatus achieves. In 1969, John Clauser, Michael Horne, Abner Shimony and Richard Holt derived a version suited to the lab, now called the CHSH inequality after their initials. The CHSH inequality uses two settings per side and sets a local limit of 2 on a combination of correlations, while quantum mechanics allows values up to about 2.8. A result clearly above 2 rules out local hidden variables, provided the experiment closes the escape routes discussed in the next level.

Bell admired Einstein and found nonlocality troubling, but he insisted the question be faced squarely, and he kept writing clear, witty papers on it until his sudden death in 1990.

Worked example: playing ten thousand rounds

Imagine running the game with real entangled photons. After ten thousand pairs, the operators meet and compare notes. In about a third of the runs they happened to pick the same setting, and in every one of those the colors matched. In the other two thirds the settings differed, and in those the colors matched only about a quarter of the time. Add it up: one third of the runs agree for certain, and a quarter of the remaining two thirds agree, which comes to one half overall.

A defender of instruction cards would need at least about 5,556 agreements out of 10,000, the five ninths minimum. The photons give about 5,000. A gap of more than five hundred is far too large to blame on chance with that many runs. The cards lose. The experiment cannot say which assumption failed, only that particles quietly carrying their answers cannot be the whole story.