Appendix 1: The GHZ Theorem

It is a complicated matter even to write down Bell’s Theorem—the particular mathematical relation between the various quantities that he proved every hidden-variable theory must obey. And the proof of his theorem is harder still. But some time after Bell’s work, an extension of his theorem was discovered—an extension so simple that it is actually possible to describe the theorem, and to give a nontechnical proof.

The authors of this new theorem are Daniel Greenberger, Michael Horne, and Anton Zeilinger—hence their result’s name: the GHZ theorem. It is an extension of Bell’s work. Recall that Bell had envisaged two particles: in the GHZ analysis there are three. Similarly, the old argument had envisaged two experimenters, Alice and Bob, while the new one has a third—Chris, let’s say. In both theorems the experimenters measure the spins of the particles heading toward them and, as with all quantum measurements of spin, there are only two possible results: the spin is found to lie either along the reference direction of the detector, or against it.

In Bell’s scenario the particles were in a special entangled state in which Alice and Bob’s measurements yielded opposite results if the detectors were parallel. So in this configuration Alice was able to predict the result of Bob’s measurement prior to his making it—it would be the opposite of what she had obtained. Recall the central point of the old EPR argument that had so disturbed Bohr: Einstein and his coworkers claimed that this demonstrated that the spin of Bob’s particle must have existed all along, contrary to the principles of quantum mechanics.

Similarly, in the Greenberger, Horne, and Zeilinger scenario the particles are also in an entangled state, so the results of measurements can also be predicted and the same conclusion would follow. And just as Bell had found a way to experimentally test the EPR conclusion, so too do Greenberger and colleagues. Thus experimental metaphysics.

The GHZ scenario goes as follows. Alice, Bob, and Chris measure the components of spin of the particles entering their detectors, and each of them writes down on a slip of paper the result. They do so adopting a particular shorthand:

Finally, the three experimenters collect their results and multiply them together. The result is itself a single number: let us call it their final combined result. What might this combined result be? It can only be plus or minus one, since each of its three individual components were either plus or minus one.

Greenberger, Horne, and Zeilinger envisage measurements in which the experimenters orient their detectors so that one is horizontal, and the other two vertical. In this particular configuration their entangled state has an important property: the final combined result can only be plus one. It can never be minus one.

Follow now the Einstein, Podolsky, and Rosen argument and see what it makes of this. Imagine with them that Alice moves far off into the distance, so that Bob and Chris make their measurements before she makes hers. Then they can predict the result she will get! For suppose that Bob had obtained, say, spin along the reference direction, and so jotted down +1, while Chris had found spin against it and so written –1. Then, since the product of all three results must be +1, they know that, when Alice makes her measurement, she is sure to obtain –1. So Bob and Chris have determined the spin of Alice’s particle: its spin is against her axis.

The same is true for the various other results Bob and Chris might have obtained. So the usual EPR argument leads to the conclusion that the hidden variable corresponding to Alice’s measurement exists.

Greenberger, Horne, and Zeilinger realized that they could test this conclusion. They would do so as follows.

Notice that there are three different ways in which Alice, Bob, and Chris can orient their detectors. Two are vertical, and only one of them horizontal—but which is the horizontal one? It might be Alice’s. Alternatively, it might be Bob’s that is horizontal, or finally Chris’s. So there are three possible cases.

In the first case we can write the final combined result as

Ahorizontal Bvertical Cvertical

where by “A” we mean the number Alice wrote down, “B” the number Bob wrote down, and so too for “C.”

In the second case the final result is

Avertical Bhorizontal Cvertical

And in the third

Avertical Bvertical Chorizontal

Each of these expressions is the product of three things—the three results that Alice, Bob, and Chris had jotted down on their slips of paper. Recall that these results (plus or minus one) are just shorthand ways to express the fact that the electron’s spin lay along or against the direction of their detectors. And recall that the usual EPR argument was that they must have been real properties of the electrons. If this is so—and here is the hidden-variable hypothesis in action—then we can treat these three results as simple numbers, and we can rearrange them in any way we wish. That is what Greenberger and coworkers proceeded to do.

They knew that in their special entangled state each of these products equals plus one. So if they were to multiply them together they would still get plus one:

(Ahorizontal Bvertical Cvertical) (Avertical Bhorizontal Cvertical) (Avertical Bvertical Chorizontal) = +1

They rearranged this:

Ahorizontal Bhorizontal Chorizontal (Avertical)2 (Bvertical)2 (Cvertical)2 = +1

And they recalled that no matter what the three experimenters obtained, the numbers they wrote down—A, B, and C—could only be plus or minus one. But the square of minus one is plus one, and so of course is the square of plus one. So they realized that they could simply leave out the terms involving the parentheses—they were just plus one. And they got a most interesting result:

Ahorizontal Bhorizontal Chorizontal = +1

In words: if all three experimenters were to orient their detectors horizontally, their final combined result is guaranteed to be plus one. This is a prediction of the hidden-variables theory, and it can be tested.

There’s more. Greenberger, Horne, and Zeilinger realized that quantum mechanics makes the opposite prediction:

Ahorizontal Bhorizontal Chorizontal = −1

So the final combined result of three horizontal measurements is sure to be minus one. So, just as in Bell’s analysis, quantum mechanics disagrees with the postulate of hidden variables. Here too, an experiment to measure this result of this configuration would have metaphysical implications.

That experiment has been done. Its results agree with quantum mechanics, and not with the hidden-variable hypothesis.