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:

- A(
*a,b,s*): the output of detector A, as a function of the detector settings and the initial quantum state at the source (we assume the output can be characterized as some kind of mathematical object, but we make no constraints on what kind); - B(
*a,b,s*): the output of detector B, as a function of the detector settings and the initial quantum state at the source (again, with no constraints on what kind of mathematical object characterizes the output); and - C(
*a,b,s*): the*correlation*between the two detector readings, as a function of the detector settings and the initial quantum state at the source (this is to be taken as a statistical "correlation coefficient" in the usual sense, so a value of 1 indicates perfect correlation between the readings, -1 indicates perfect anticorrelation--the readings are always "opposite", in whatever sense is appropriate for the particular types of detectors involved--and 0 indicates no correlation--the readings are statistically independent).

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:

- A(
*a,b,s*) = A(*a,s*), - B(
*a,b,s*) = B(*b,s*),

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:

- A(
*a,s*) = average over all*h*of A(*a,s,h*), - B(
*b,s*) = average over all*h*of B(*b,s,h*)

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:

- C(
*a,b,s*) = f(*a,s*) * g(*b,s*),

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.