In 1935 Einstein, Podolsky and Rosen pointed out that Alice, by measuring her particle along z, can predict Bob's result along z with certainty, without touching Bob's particle. They concluded that Bob's result must have been fixed all along, and that quantum mechanics, which gives only probabilities, leaves something out.
Their view can be turned into a model. When a pair is made, each particle is given a list of answers, one for every axis it might be measured along, with Bob's list the opposite of Alice's. A measurement only reads the answer from the list. Because the lists are fixed at the source and each particle has its own, such a model is called a local hidden variable model.
The model explains what the article on entangled spins found along a single axis. Along the same axis the results are always opposite, because the lists are opposite. Each side alone sees half and half, if the lists are made at random. And no message can be sent, because nothing that Alice does changes Bob's list.
For almost thirty years it seemed that no experiment could decide between this view and quantum mechanics. In 1964 John Bell showed that the two make different predictions when the two sides measure along different axes.
Bell's argument is easiest to follow in the form found in 1969 by Clauser, Horne, Shimony and Holt, called CHSH. Alice measures along one of two axes, a or a', and Bob along b or b', each choosing at random for each pair. Every result is +1 or -1, and E(x, y) is the average of the product of the two results when the axes are x and y.
The four averages are combined into one number S, the sum of E(a,b) - E(a,b') and E(a',b) + E(a',b'). In a hidden variable model each pair has fixed answers A and A' for Alice's two axes and B and B' for Bob's. For one pair, the four products combine into A(B - B') + A'(B + B').
Since B and B' are each +1 or -1, one of B - B' and B + B' is 0 and the other is \pm 2. The expression is therefore +2 or -2 for every pair, whatever the lists are. An average of numbers that are each \pm 2 lies between -2 and 2, so |S| \le 2.
This is the CHSH inequality. It rests on two assumptions only: that each result is fixed before the measurement, and that the result on one side does not depend on the axis chosen on the other side.
For the singlet, the correlation between results along two axes at an angle \theta is E = -\cos\theta. Placing the four axes in equal steps of \varphi, with a, b, a' and b' at 0, \varphi, 2\varphi and 3\varphi, gives |S| = |3\cos\varphi - \cos 3\varphi|.
At \varphi = 45^\circ, with a = 0^\circ, b = 45^\circ, a' = 90^\circ and b' = 135^\circ, this gives |S| = 2\sqrt{2} \approx 2.83, well above the limit of 2. No quantum state gives more than this; Boris Tsirelson proved that upper limit in 1980.
For photons the angles are halved, so the polarizers are set at 0^\circ, 22.5^\circ, 45^\circ and 67.5^\circ.
The difference between 2 and 2.83 is large enough to measure. If it is found, then no list of answers attached to the particles can give the observed correlations, as long as the two sides cannot affect each other.
The first tests, by Freedman and Clauser in 1972, used photon pairs from calcium atoms and found a violation of a Bell inequality. In 1982 Alain Aspect and his colleagues measured S = 2.697 \pm 0.015. In another experiment that year they changed the settings of the polarizers while the photons were on their way, so that the setting on one side could not be known on the other side in time.
In 1998 Weihs and his colleagues placed the two stations 400 m apart, chose each setting at random just before detection, and measured S = 2.73 \pm 0.02. In 2015 three groups, in Delft, Vienna and Boulder, did experiments in which the settings were chosen too late for any signal to pass between the stations, and in which so many pairs were detected that the undetected ones could not change the conclusion. The Delft group used electrons in two diamonds 1.3 km apart and measured S = 2.42 \pm 0.20.
None of these experiments found a value above 2\sqrt{2}, and all of them found values above 2. In 2022 the Nobel Prize in Physics was awarded to Aspect, Clauser and Anton Zeilinger for experiments with entangled photons.
The conclusion is that the results are not fixed in advance by a list that each particle has, if the choice of setting on one side cannot affect the other side. Quantum mechanics still does not allow these correlations to be used to send a message.
In 1991 Artur Ekert proposed using entangled pairs to give two people a shared secret key. Alice and Bob measure every pair along an axis chosen at random from three, Alice from 0^\circ, 45^\circ and 90^\circ and Bob from 45^\circ, 90^\circ and 135^\circ. Afterward they announce which axes they used, but not their results.
Pairs measured along the same axis give opposite results, so Bob reverses each of his bits from those pairs, and both then hold the same random string. That string is the key. Nobody else knows it, because the results did not exist before the measurements.
The pairs measured along different axes are used to compute S. If someone had intercepted the particles and measured them, or had prepared the pairs so as to know the results in advance, the results would follow a list fixed beforehand and S could not exceed 2. A value close to 2\sqrt{2} shows that no one else has the results.
Keys of this kind have been shared over optical fibers and, in 2020, between two ground stations 1{,}120 km apart using the Micius satellite. The inequality that Bell derived to compare quantum mechanics with hidden variable models is now used to check that a key is private.