EPR, Bell's Inequalities, Nonlocality, And All That

Ever since the advent of quantum mechanics, there has been a significant cottage industry in philosophical and metaphysical speculations about "what it all means", what it implies about "the nature of reality", what it says about how our minds "interact with reality", and so forth. Many if not most of these speculations have led nowhere useful, of course, but here I want to talk about one aspect of the subject that actually is worth some deep thought. (If you've read my review of The Dancing Wu Li Masters, you'll know that I don't have much patience with people trying to read deep metaphysical implications into what are actually pretty prosaic phenomena, but even by that standard, the stuff I'm going to talk about in this article is pretty weird, as I admit in that review, and deserves some discussion--not that Zukav's discussion is really all that helpful once he gets beyond the bare description of the phenomena themselves, as I also say in the review.)

Throughout the early years of quantum mechanics, Einstein had a long-running debate with Bohr over whether the theory really gave a complete description of reality. Einstein would come up with a thought experiment that seemed to show that there were aspects of reality not captured in the quantum mechanical description, and Bohr would figure out some loophole in the argument that made it seem that quantum mechanics was vindicated after all. One such debate occurred in 1935, when Einstein published a paper with two other physicists, Boris Podolsky and Nathan Rosen (hence "EPR") that described a new thought experiment, and Bohr, after consideration, came out with a reply that once again seemed to leave quantum mechanics triumphant. (I'll describe the basic outline of the EPR argument below.)

During all this, of course, nobody had any expectation that these "thought experiments" would ever turn into real experiments that could actually be run. Einstein had always been big on thought experiments, but he, like most physicists, understood them to be simply exercises in logic and imagination--attempts to bring into sharp focus some particular issue, to explore the consequences of some key assumption or implication of a theory, but certainly not intended to lead to actual empirical tests. However, in 1964, John Bell published a paper which took the EPR thought experiment and turned it into something which could potentially be run as a real experiment--and what's more, he derived certain inequalities which could be used to examine the data from such an experiment and actually distinguish, in the real world, the predictions of standard quantum mechanics from those which would follow from the view set forth in the EPR paper. Since that time, actual experiments have been run along these lines, and (not surprisingly to most physicists) the results have confirmed the predictions of standard quantum mechanics and ruled out the kind of model of reality that EPR argued for.

You'd think that would be the end of this chapter in the history of quantum mechanics, but au contraire. The cottage industry I referred to above, which deals in philosophical and metaphysical speculations about quantum mechanics and what it means, has been working overtime to try to find some way around the obvious conclusions that follow from Bell's paper combined with the experimental results. The experiments themselves have been criticized on the grounds that the measurements are too inaccurate to permit drawing any conclusions (and it's true that some of the apparatus used, such as "detectors" for the various types of particles involved, are not very sensitive by everyday standards, so there is a lot of data that basically just says "no answer"--but as Bell said in one of his papers on the subject, given that the data from these insensitive detectors, with standard statistical corrections applied to it, confirms the quantum mechanical predictions so well, it's hard to see how having more accurate detectors could change the results drastically enough to matter). And the philosophers have had a field day looking for hidden assumptions in Bell's arguments and deciding which one they want to try to deny in order to come up with some interpretation of what's going on that they can live with.

So why all the ruckus? Here I want to give a simple look at what's going on and what it seems to imply, and why so many people find that obvious implication so hard to accept. The description I give of the experimental situations in question will be very schematic, and the path I take from that to the conclusions that have people so exercised will be as short and simple as I can make it, so you can see how few assumptions are really involved (and I will try to make all of those assumptions explicit, so you can see exactly what's at stake). This is very much in the spirit of Bell's own work, and my aim here is much the same as his was: to cut through all the philosophical smoke and mirrors and find a simple picture that the man in the street can deal with.

We start with the schematic description of the kind of experimental setup we'll be dealing with. We have a source S which spits out pairs of quantum "particles" (what kind is immaterial--they could be electrons, photons, anything--and I put the word "particles" in quotes because I want to make it clear that we're actually making no assumptions about what kind of quantum systems we're dealing with, just so they obey the rules of quantum mechanics). We set things up so that each pair of "particles" emitted by the source is in the same joint quantum state s, where "joint" quantum state means that we do not necessarily specify individual states for the two "particles" in isolation--we only guarantee that there is a single state s that describes the pair of particles as a single system. (Again, we make no assumptions about how the quantum state is specified, what kind of mathematical object it is, and so on, just so it obeys the rules of quantum mechanics.)

The two "particles" in each pair then travel in opposite directions to detectors A and B, which "measure" some property of the "particles" (again, "measure" is in quotes because we're making no assumptions about what kinds of detectors, what property they're measuring, what kinds of interactions they use to make the measurements, and so on, just so, once again, they obey the rules of quantum mechanics). We set things up so the two detection events, A and B, are spacelike separated (which just means the locations and times of the detection events are such that there is not enough time for light to travel between them); this requirement will become important in what follows, when we try to square what's going on with relativity.

We then run this experiment a large number of times and analyze the data to see what interesting patterns we can find. In particular, we can look at the statistics of the detector readings at A and B; we can look at each detector's reading in isolation, and we can look at correlations between the readings of the two detectors. We can also vary the "settings" of the detectors in various ways and see how that affects what we measure. We can then check how all of these results match up with what we would predict from our knowledge of the initial quantum state s of the two-particle system.

Let's adopt some more terminology to make things easier to follow. We'll let a stand for the "settings" of detector A, and b for those of detector B. (Once again, we aren't making any assumptions about what these "settings" are, except that we can specify them in such a way that we can use them as "independent variables" with which to plot statistics. What that means will become clear in a moment.) We can then look at the following:

Here we come to the first issue raised in the EPR paper. Since the detection events at A and B are spacelike separated, it ought to be the case that the output of detector A does not depend on b (the settings of detector B), and vice versa. So we ought to be able to look at our data and show the following:

or, in other words, the output of detector A is statistically independent of the settings of detector B, and vice versa. This is called the "locality assumption". It seems obvious (and indeed required if one's theory is to be consistent with special relativity), but, as we shall see, things are not so simple.

Next, you will note that I have included the source state s as one of the "independent variables" in our statistical plots above, even though we're not going to vary it. Why? This is the second issue raised by the EPR paper, which I'll state in a slightly different form than theirs (a form due to David Bohm in a 1952 paper, on which Bell drew heavily for his own 1964 paper). It turns out that we can set up our source to produce particle pairs in a certain quantum state s which has the following interesting property: if the settings a and b for the two detectors are the same, then the readings are perfectly anticorrelated: that is, for any given detector setting a, C(a,a,s) = -1. Note that this holds true regardless of what the actual detector setting a is, as long as it's the same for both detectors. But more importantly, this perfect anticorrelation holds true even though the readings of the individual detectors are perfectly random--that is, there is no way to predict, just from knowledge of the initial state s and the detector setting a, what the outputs A(a,s) and B(a,s) are going to be for any given run of the experiment (other than the fact that they'll always be opposite if the detector settings are the same). That is, although we can use the rules of quantum mechanics to predict the statistical distribution of the results of a large number of runs, each individual run, as far as we can tell from the quantum rules, gives a completely random result "drawn" from that distribution (like a single random card drawn from a deck of cards).

The EPR argument now puts all the above together to come up with the following conclusion: since the output of detector A can't depend on the detector setting at B, and the output of detector B can't depend on the setting at A, the only way to ensure perfect anticorrelation if the settings are the same is for the outputs to be determined in advance for both particles for that setting. But this holds for any detector setting at all, so in fact the outputs of both detectors must be determined in advance for all possible runs of the experiment. Yet we can't predict those outputs just from knowledge of the quantum state s, as we saw above. EPR's conclusion: the quantum state cannot be a complete specification of the "real" state of the particles. In other words, there must be some extra variables, other than the quantum state s, for the detector outputs A and B and the correlation function C to depend on, and it's the values of those extra variables, which we'll call h, for "hidden variables", since that's what they're usually called in the literature, that will determine the actual detector outputs A and B for a given run of our experiment. That is, we will have:

for the statistics of our detector outputs over a large number of runs. (In general we will not actually know the values of our hidden variables h for any given run of our experiment, but we're assuming that we know enough about their statistical distribution to compute the averages above.)

The EPR argument stops at this point, with the (apparent) demonstration that quantum mechanics must be incomplete. I won't go into Bohr's reply here because, although it was accepted as refuting EPR's demonstration, it didn't really address the issues we're discussing, instead going off on a completely different tangent--which I will say is rather surprising in hindsight, given what we are about to see from Bell's paper. For almost 30 years after that 1935 discussion nobody seems to have asked the question that Bell asked in 1964, and yet the question seems perfectly obvious once asked.

Bell's question was a simple one: is the "locality assumption" we made above, which seemed so obvious, consistent with the predictions of quantum mechanics? Surprisingly, he found that the answer is no! That is, quantum mechanics predicts that the statistics for detector A's output will not be independent of the settings of detector B, and vice versa! Bell's proof was straightforward: if the locality assumption is true, then the correlation function C will have to factorize as follows:

where f and g are functions of their respective detector settings and the initial quantum state s. (Note that this does not rule out the possibility that f and g also depend on other "hidden" variables h--if they do, the above equation must still hold, when f and g are averaged over all possible values of h.) However, Bell was able to prove mathematically that the actual correlation function C predicted by quantum mechanics cannot be factorized in this way! He did this by showing that any function which can be factorized as above must obey certain inequalities (the "Bell inequalities") which are violated by the quantum mechanical correlation function C. (Again, I stress that this conclusion holds even if C also depends on additional "hidden variables".)

The Bell inequalities are important because, for the first time, they gave experimenters something they could actually test. The original EPR argument, and its subsequent versions by Bohm and others, didn't make any actual numerical predictions about the statistical distributions A, B, and C above. Bell did, and it was not long before experimenters began to take up the challenge. As I noted above, to date all the evidence indicates that the Bell inequalities are violated, and the predictions of quantum mechanics are correct. This leads us to a conclusion that many people find unpalatable: the "locality assumption" above cannot be true. The output of each detector A and B in our experiment has to depend on the settings of both detectors, even though the detection events are spacelike separated. That means, for example, that (as has now been done in several experiments) we can set things up so that the actual settings for each detector are only chosen after the pair of particles has left the source, each headed for its own detector--meaning that whatever is happening to produce the quantum correlations between the detector readings would seem to be happening "faster than light".

I'll make several quick observations at the end here, just as comments on the vast literature produced by the cottage industry I referred to above. You will note that Bell's conclusion is a very simple and direct one: "locality" cannot be true; "nonlocality" must be a feature of quantum phenomena. It doesn't matter whether or not the quantum state is a "complete" description of reality or whether it has to be supplemented with "hidden variables". It doesn't matter whether quantum systems are "real" before they are observed or measured. We didn't make any assumptions or commitments about any of those things above. It doesn't matter how "counterfactuals" about measurements that we could have made but didn't are handled; all of our statements above were made in terms of actual observed statistics. (We did have to assume that we could in fact set up our source to ensure that each pair of particles is in the same joint quantum state, but that assumption is a straightforward empirical one with no "counterfactuals" in it, and it certainly seems borne out by experiment.) All we did was to observe that, if "locality" were true, our observed correlation function between the two detectors would have a certain mathematical property which, as a matter of both quantum mechanical prediction and (now) experimental observation, it doesn't have. No obvious loopholes there--not that I expect the cottage industry to die down any time soon.


There are a lot of discussions of this topic around, and the vast majority of them are at best ill-informed, and at worst not even wrong (to use Pauli's pithy phrase). (The vast majority of them are by amateurs, which may have something to do with it--but of course I'm not a specialist in this area either, so I would not want to say that all amateur discussions are worthless.) Rather than try to wade through even a smattering of all this, I've limited myself to two links that cover the essential ground and provide further references for your own explorations.

Bell's Theorem at Wikipedia: Good general info, with more specific examples of the kinds of quantum systems that violate the Bell Inequalities, and the kinds of experiments that have been done to test them.

Does Bell's Inequality Principle rule out local theories of quantum mechanics?: A link on the Usenet Physics FAQ site. Short and simple but gets across the main points.

As far as actual book references go, I can do no better than to recommend John Bell's own collection of papers on the subject, Speakable and Unspeakable in Quantum Mechanics (it's available at Amazon). In addition to the specific issues discussed in this article, Bell's book also has a lot of good discussion on the more general philosophical aspects of quantum mechanics.