In a famous paper published in 1935, Einstein, Boris Podolsky (1896–1966), and Nathan Rosen (1909–95) argued that, if the predictions of quantum mechanics about the outcomes of experiments are correct, then the quantum mechanical description of the world is necessarily incomplete.

A description of the world is “complete,” according to the authors (EPR), just in case it leaves out nothing that is true about the world—nothing that is an “element of the reality” of the world. This entails that one cannot determine whether a certain description of the world is complete without first finding out what all the elements of the reality of the world are. Although EPR do not offer any method of doing that, they do provide a criterion for determining whether a measurable property of a physical system at a certain moment is an element of the reality of the system at that moment:

If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that physical quantity.

This condition has come to be known as the “criterion of reality.”

Suppose that someone proposes to measure a particular observable property P of a particular physical system S at a certain future time T. Suppose further that there is a method whereby one could determine with certainty, prior to T, what the outcome of the measurement would be, without causing any physical disturbance of S whatsoever. Then, according to EPR, there must now be some matter of fact about S—some element of reality about S—by virtue of which the future measurement will come out in this way.

EPR’s argument involves a certain physically possible state of a pair of electrons that has since come to be referred to in the literature as a “singlet” state or an “EPR” state. Whenever a pair of electrons is in an EPR state, the standard version of quantum mechanics entails that the value of the x-spin of each electron will be equal and opposite to the value of the x-spin of the other, and likewise for the values of the y-spins of the two electrons.

Assume that there is no such thing as action at a distance: nothing that happens in one place can cause anything to happen in another place without mediation—without the occurrence of a series of events at contiguous points between the first location and the second. (Thus, the flipping of a switch in one room can cause the lights to come on in another room, but not without the occurrence of a series of events consisting of the propagation of an electric current through a wire.) If this assumption of “locality” is true, then it must be possible to design a situation in which the pair of electrons in the ERP state cannot interact with each other and in which, therefore, any measurement of one electron would cause no disturbance to the other. For example, the electrons could be separated by a great distance, or an impenetrable wall could be inserted between them.

Suppose then that a pair of electrons in an EPR state, e₁ and e₂, are placed at an immense distance from each other. Because the electrons are in an EPR state, the x-spin of e₁ will always be equal and opposite to the x-spin of e₂, and the y-spin of e₁ will always be equal and opposite to the y-spin of e₂. Then there must be a means of determining, with certainty, the value of the x-spin of e₂ at some future time T without causing a disturbance to e₂—namely, by measuring the x-spin of e₁ at T. Likewise, it must be possible to determine with certainty the value of the y-spin of e₂ at T, without causing a disturbance to e₂, by measuring the y-spin of e₁ at T. By the criterion of reality above, therefore, there is an “element of reality” corresponding to the x-spin and y-spin of e₂ at T; that is, there is a matter of fact about what the values of e₂’s x-spin and y-spin are. But, as discussed earlier, it is a feature of the standard version of quantum mechanics that it is impossible to determine the simultaneous values of the x-spin and y-spin of a single electron, because the measurement of one always uncontrollably disrupts the other (see above The principle of superposition). Hence, the standard version of quantum mechanics is incomplete. Parallel arguments can be constructed by using other pairs of distinct but mutually incompatible observable properties of electrons, of which there are literally an infinite number.

If the existence of an EPR state entails an infinity of distinct and mutually incompatible observable properties of the electrons in the pair, then the statement that the EPR state obtains—because the EPR state does not specify a value for any such property—necessarily constitutes a very incomplete description of the state of the pair of electrons. The statement is compatible with an infinity of different “true” states of such a pair, in each of which the observable properties assume a distinct combination of values.

Nevertheless, the information that the EPR state obtains must certainly constrain the true state of a pair of electrons in a number of ways, since the outcomes of spin measurements on such pairs of electrons are determined by what their true states are. Consider what sorts of constraints arise. First of all, if the EPR state obtains, then the outcome of a measurement of any of the above-mentioned observable properties of e₁ will necessarily be equal and opposite to the outcome of any measurement of the same observable property of e₂. In other words, whenever the EPR state obtains, the true state of the pair of electrons in question is constrained, with certainty, to be one in which the value of every such observable property of e₁ is the equal and opposite of the value of the same observable property of e₂.

There are statistical sorts of constraints as well. There are, in particular, three observable properties of these electrons—one of them is the x-spin, and the others may be called the k-spin and the l-spin—that are such that, if any one of them is measured on e₁ and any other on e₂, the values will be opposite one-fourth of the time and equal three-fourths of the time.

At this point a well-defined question can be posed as to whether these two constraints—the deterministic constraint about the values of identical observable properties and the statistical constraint about the values of different observable properties—are mathematically consistent with each other. In 1964, 29 years after the publication of the EPR argument, the British physicist John Bell showed that the answer to this question is “no.”

Thus, the EPR state implies a mathematical contradiction. The conclusion of the EPR argument, therefore, is logically impossible. It follows that one of the two assumptions on which the EPR argument depends—that locality is true (there is no action at a distance) and that the predictions of quantum mechanics regarding spin measurements on EPR states are correct—must be false. Since the predictions of quantum mechanics regarding spin measurements are now experimentally known to be true, there must be a genuine nonlocality in the workings of the universe. Bell’s conclusion, now known as Bell’s inequality or Bell’s theorem, amounts to a proof that nonlocality is a necessary feature of quantum mechanics—unless, which at this writing seems unlikely, one of the “many worlds” interpretations of quantum mechanics should turn out to be correct (see above The theory of Ghirardi, Rimini, and Weber).