In the first instalment of our series on the Foundations of Quantum Mechanics, Tim Maudlin revisits Einstein’s famous disquiet with a prediction of quantum mechanics: the possibility of "spooky action at a distance". As John Bell later proved, quantum mechanics is indeed a non-local theory, allowing for the outcomes of experiments separated by enormous distances to causally influence each other.
It seems to be a manifest fact about the physical world that in order for one action or event to have an effect on another there must be some continuous process—an exchange of particles, flow of electricity, flash of light, etc—that connects them. If such continuous processes are required and if those processes are in turn limited by the speed of light, then there would be a strict limitation on what can have a causal effect on what. Indeed, denying this sort of limitation on causation is what Albert Einstein famously referred to as “spooky action-at-a-distance” or “telepathy”, and he roundly rejected the possibility. But quantum mechanics shows that, despite Einstein's protestations, the physical world is non-local.
In 1935, Einstein, Nathan Rosen and Boris Podolsky published a paper which would—in a way they could not possibly anticipate—result in arguably the most astonishing result in the history of physics. The point of argument was not to suggest any new and remarkable physical phenomenon. Indeed, the only observable phenomena mentioned in the paper were commonplace and intrinsically uninteresting sorts of correlations which can be trivially explained. The point that EPR (as the three authors are now referred to) were making is that although the correlations can be trivially explained in a pedestrian way, the standard understanding of quantum mechanics does not so explain them. Indeed, the fundamental principles of quantum mechanics as exposited by Neils Bohr and the Copenhagen school forbids explaining them in the obvious trivial way. EPR presented the phenomena not for their intrinsic interest but as a vivid illustration of some bizarre consequences of the Copenhagen approach.
The phenomena that EPR mention are perfect correlations predicted by quantum theory when particular experiments are carried out on separated pairs of particles. Following the standard rules of the quantum formalism, EPR write down a so-called entangled quantum state for the pair. One of the particles is sent off to Alice and the other to Bob, located in arbitrarily distant labs. Alice and Bob each carry out the procedure called a “momentum measurement” on their particle, and since the entangled state is a zero momentum state for the pair, the prediction is that whatever value Alice finds, Bob will obtain the opposite. In this sense, the outcomes of the experiments will be perfectly negatively correlated. Alice and Bob can accurately predict the outcome of the other’s experiment once they have carried out their own.
These sorts of perfect correlations—and the predictions they underwrite—are in principle trivial to explain. Take a dollar bill, tear it in half, randomly mix up the two halves and then send one to Alice and the other to Bob. Explain the protocol to Alice and Bob. Then before opening their envelopes neither Alice nor Bob can predict what they will find, but once either of them has looked he or she will be in a position to predict with perfect accuracy what the other will find. Or—to take an example John Bell uses—once one knows that Professor Bertlmann always wears different colored socks, seeing the color of one will allow one to predict with perfect accuracy something about the color of the other: it will be different. Big deal. All of this should seem quite everyday and uninteresting because it is.
The entangled quantum state used to make the EPR predictions has another characteristic: although it allows one to predict the anti-correlation between the momenta with certainty, it does not allow one to predict what the exact value of the momentum will be. That characteristic is shared by our everyday examples: if all one knows is that one of the two halves of the dollar has been put in each envelope, or that Bertlmann always wears different colored socks, then one can predict the anti-correlation of the outcomes of Alice and Bob’s experiments without being able to predict the exact outcome in either case. But it is essential to making the physical explanation trivial that one assumes that the outcomes are already predetermined by the initial preparation of the two systems. Already when they are put in the envelopes, one specific half of the dollar bill went in Alice’s and the other went in Bob’s. Already when Bertlmann got dressed in the morning, the sock on his right foot was some specific color and the sock on his other foot was a different specific color. Alice and Bob might not know the specific halves of the bills in their envelopes, or the specific colors Bertlmann chose, but some specific half was put in each envelope and some specific color sock went on each foot. So Alice’s and Bob’s information about the initial physical situation is incomplete: it leaves out specific physical details of the envelopes and socks. Under the assumption of this sort of incompleteness of the initial descriptions, the observed perfect anti-correlation is not puzzling or problematic at all.
The critical observation that EPR made is that according to the standard Copenhagen understanding of quantum theory, before Alice does her experiment, her particle has no specific momentum. Nor does Bob’s. Nonetheless, the perfect anti-correlation can be predicted—accurately!—from the quantum state.
The idea that observation somehow indeterministically creates the reality of what is observed is often associated with quantum theory, and is in itself peculiar enough.
Although their paper is often described as a “paradox”, that is not how EPR regarded it. They presented it as a straightforward argument that the quantum-mechanical description of a system like the two particlescannot be complete exactly because it predicts the perfect anticorrelation between the results without predicting either exact result. As Bell put it (citing Max Jammer’s account of Pascual Jorden):
‘Jordan declared, with emphasis, that observations not only disturb what has to be measured, they produce it. In a measurement of position, for example, as performed with the gamma ray microscope, “the electron is forced to a decision. We compel it to assume a definite position; previously it was neither here nor there; it had no yet made its decision for a definite position….If by another experiment the velocity of the electron is being measured, this means: the electron is compelled to decide itself for some exactly defined value of the velocity…we ourselves produce the results of measurement”’.
It is in the context of ideas like these that one must envisage the discussion of the Einstein-Podolsky-Rosen correlations. Then it is a little less unintelligible that the EPR paper caused such a fuss, and that the dust has not settled even now. It is as if we had come to deny the reality of Bertlmann’s socks, or at least of their colours, when not looked at. And as if a child has asked: How come they choose different colours when they are looked at? How does the second sock know what the first sock has done? 
The idea that observation somehow indeterministically creates the reality of what is observed is often associated with quantum theory, and is in itself peculiar enough. But what the EPR paper pointed out is that for certain distantly separated systems with an appropriately entangled wavefunction the problem becomes much worse. For to maintain the perfect correlations (or perfect anticorrelations) between the outcomes in such a setting, the (indeterminstic) outcome of one experiment must somehow be instantaneously communicated to the other particle, else its behavior cannot be guaranteed to display the correlation. As Einstein later put it:
“It seems hard to sneak a look at God’s cards. But that he plays dice and uses ‘telepathic’ methods (as the present quantum theory requires of him) is something that I cannot believe for a single moment.”
Note that the “God plays dice” (indeterminism) part is linked directly to the “spooky action-at-a-distance” (telepathy) part of his complaint. It is precisely the perfect EPR correlations between distant outcomes that secures that linkage. And as we know from other letters, it was the spooky action-at-a-distance part that Einstein really found unacceptable, not least because it appears to be incompatible with Relativity.
Then EPR’s point is twofold. One is that the correlations they point out can clearly be accounted for by a local theory. And the other is that quantum theory as exposited by the Copenhagen school is not local.
Call a theory with no spooky action-at-a-distance—no telepathy, no way for information about what happens in one lab to be instantly transmitted to (and have an effect in) the other—a local theory. Then EPR’s point is twofold. One is that the correlations they point out can clearly be accounted for by a local theory. And the other is that quantum theory as exposited by the Copenhagen school is not local. If the quantum wavefunction is complete, then the pedestrian explanation of the correlations is barred, because the particular outcomes in each lab cannot be inferred from the wavefunction. If the wavefunction is complete, then the outcome on at least one side must be indeterministic, and the outcome of that indeterministic experiment must somehow constrain what happens on the other side. Such a theory is then non-local.
Finally, note that EPR’s point is not that determinism per se is somehow required or even desirable. Rather, it is locality that is demanded, and local determinism is the only possible way to achieve it in the face of the correlations. Bell again:
It is important to note that to the limited degree to which determinism plays a role in the EPR argument it is not assumed but inferred. What is held sacred is the principle of “local causality”—or “no action at a distance”. 
So what did Bell himself contribute to this situation? His famous 1964 paper “On the Einstein-Podolosky-Rosen paradox” takes up exactly where EPR left off. If there are perfect correlations—correlations that allow Bob to infer from his observed result what Alice’s result must be—then any local theory must also be deterministic. If you want to preserve locality, determinism is the only game in town. What Bell realized is that if one looks at the predictions that quantum mechanics makes, no local deterministic theory can make the same predictions and therefore—given the EPR result—no local theory at all can make the same predictions. It is locality itself that is incompatible with the phenomena. So at the end, Einstein loses and spooky action-at-a-distance wins.
Bell’s own example, like EPR, involved pairs of entangled particles and experiments done on their spins. But for expository purposes, the point is most clearly made by an example later developed by Daniel Greenberger, Michael Horne and Anton Zeilinger (as exposited by David Mermin).  This later example uses a triple of entangled particles, which are sent off to Alice, Bob, and Charlie in arbitrarily widely separated labs. The separation ensures that in a local theory, Alice’s, Bob’s, and Charlie’s decision about how to arrange their experimental apparatus cannot be communicated to—or have any influence on—the way the particles in other labs interact with their apparatuses or what the outcomes of those experiments are.
What we now know—due to Bell’s proof and the experimental tests—is that the physics of the actual world must somehow violate Einstein’s proscription on “spooky action-as-a-distance”.
Alice, Bob, and Charlie each can choose between two experimental arrangements, which we will call “x-spin” and “z-spin”. Practically, that means orienting a certain magnet in either the x-direction or in the orthogonal z-direction. Since there are three labs with two possible experimental arrangements each, there are a total of 8 possible global experimental conditions on each run of the experiment (after a central source delivers one of the three entangled particles to each lab). By a convenient and obvious notation we will refer to these eight possible global arrangements as XAXBXc, XAXBZc, XAZBXc, ZAXBXc, XAZBZc, ZAXBZc, ZAZBXc, and ZAZBZc. Whenever Alice, Bob, or Charlie perform an experiment on their particle they get one of two possible results. They pass the particle through the magnetic field, and it is either deflected towards the north pole of the magnet (called an “up” result) or away from it (a “down” result). As far as the experimental situation goes, that’s all you need to know.
Now for a particular prepared initial quantum state (which is referred to as an “entangled” state because it cannot be expressed as just the sum of independent separate states of the two particles), the quantum formalism makes some 100%, sure-fire, absolute predictions. These predictions play the same role in this argument as the EPR correlations play in their argument. Of course, the theory makes predictions for all eight possible experimental arrangements, but of these only four concern us: XAXBXc, XAZBZc, ZAXBZc, and ZAZBXc. Here are the predictions:
1) If all three orient their magnets in the x-direction (XAXBXc), then there will be an odd number of “up” outcomes. There might be one and there might be three, but it will certainly be odd.
2) If exactly one orients her or his magnet in the x-direction (XAZBZc, ZAXBZc, or ZAZBXc), then there will be an even number of “up” outcomes. There might be zero and there might be two, but it will certainly be even.
That’s it. These are predictions of the quantum formalism and (more importantly!) they are what actually occurs in the lab. What we now will show is that no local theory can make these predictions.
The first part of the argument just recapitulates EPR. Note that in any of these four cases the results of any two of the experiments allow one to infer with certainty the result of the third. If the total number of “up”s must be odd, for example, then given any two outcomes there is only one acceptable value for the last. From this perfect correlation, we arrive at the EPR conclusion: if the physics is local, then it must also be deterministic. For if something irreducibly chancy happens in (say) Alice’s lab, then the particles in Bob’s and/or Charlie’s must be somehow sensitive to how it came out in order to ensure the correct number of “up” outcomes. Just as in the EPR argument, this determinism is not assumed but inferred from locality and the correlations.
So if we are to maintain locality, the particles must be pre-programmed regarding how they will react to either of the two possible experimental arrangements. Because of the locality, their reaction to the experimental arrangement in their own lab cannot depend on which experimental arrangement is chosen in the distant labs. That would be spooky action-at-a-distance. Each particle must be prepared for either possible arrangement, without regard to the others.
But now we have a straightforward mathematical question: can their reactions be somehow pre-arranged at the source so that they will yield an acceptable set of outcomes no matter which of the four global experimental arrangements happens to be chosen? And the simple answer is: they cannot.
Mermin provides a beautiful graphical proof. We need to pre-arrange outcomes for each of the three particles for each of two possible experimental arrangements in their lab. Hence there are a total of six decisions to make. We will represent each decision by a circle to be filled in with either a “U” or a “D” depending on whether the result should be up or down. We arrange these six circles as follows:
Our mathematical question is now this: is there any possible way to write either “U” or “D” in each circle so that all of our correlations will be respected? That is, along the dotted line representing XAXBXc there must be an odd number of Us, and along each of the three solid lines, representing XAZBZc, ZAXBZc, and ZAZBXc, there must an even number of Us.
I commend to the reader to give it a try. And after you become sufficiently frustrated, consider the following proof that all of our four correlational constraints cannot be simultaneously met. For suppose they could. Now pick up the three disks along the dotted line and throw them in a hat. You have just thrown in an odd number of Us. Then pick up the three disks along each of the three solid lines and throw them in the hat. In each case, it is an even number of Us. So when you are done, you would have to have an odd+even+even+even = odd number of Us in the hat.
But because each disk lies at the intersection of two lines, each disk was thrown in twice. So no matter how you write “U” and “D” on the disks, the task cannot be done. The outcomes cannot be pre-arranged to come out correctly no matter which experiments are chosen. So no local theory can return these predictions. Unlike in the EPR case, determinism cannot come to the rescue of locality. Nothing can. A world that displays these phenomena must be a non-local world—must contain causal relations between arbitrarily distant events that cannot be connected even by light—, because the phenomena violate Bell’s Inequality.
The 2022 Noble Prize in Physics was awarded to John Clauser, Alan Aspect and Anton Zeilinger for experiments proving that Bell’s Inequality is, in fact, violated for similar experiments done in labs further and further apart and with more and more efficient detectors. Those experiments have ruled out locality in physics: whatever the correct physical theory is, it must somehow violate the locality condition that Bell assumed and Einstein demanded. What we now know—due to Bell’s proof and the experimental tests—is that the physics of the actual world must somehow violate Einstein’s proscription on “spooky action-as-a-distance”. It is a result as worthy as any in history of the Nobel Prize. It is a pity that Bell himself did not live long enough to receive it.
 John Bell, Speakable and Unspeakable in Quantum Mechanics, 142-3
 Letter to Cornel Lanczos (21 March 1942), p. 68 - Albert Einstein: The Human Side (1979).
 N. David Mermin, “What Wrong with These Elements of Reality?”, Physics Today, June 1990, pp. 9-11