Hidden Variables

“We now know that the moon is demonstrably not there when nobody looks.”

N. David Mermin

Last Wednesday, I was in a pub with some friends. That is to say, we are no longer colleagues, yet a good time was had by all (except for the one who got drenched in beer by the newbie waitress), so I’d say “friends” would be the appropriate classifier.

A laser pointer happened to be present, and we were playing with it, so of course the conversation turned to quantum mechanics and the Einstein-Podolsky-Rosen paradox. We remembered what the paradox was about and what it was supposed to prove, but we didn’t quite remember why it was a paradox; in other words, why the hidden variables hypothesis was not an acceptable alternative to nonlocality. So I looked it up afterwards.

Here’s a quick refresher. In the EPR experiment, a pair of quantum particles is created which are entangled: when the ‘spin’ property of both particles is measured on a given axis, they will always give opposite results. The two particles can be sent away into different directions, kilometers or even light-years apart, and they will retain their entanglement. The spin direction of each particle is unknown until measured, but as soon as you measure one of them, you know what the result of measuring the other one will be. EPR is ofter referred to as a thought experiment, but versions of the experiment can and have been performed in practice; the fact that entanglement happens is not in doubt.

The reason why the EPR paradox is so important is because it is supposed to demonstrate nonlocality: after measuring the spin of particle A, the result of measuring the spin of B is no longer uncertain but can be predicted with 100% certainty. Therefore, in effect, measuring A has influenced the measurement of B, even though they may be light years apart, which suggests that something has moved between the two particles at faster-than-light speed.

Thus far we remembered. What we were a bit hazy about, however, is this: what’s the big deal again? What’s so special about the fact that observing one object, quantum or otherwise, gives me information about another object which is far away? Why do we need all this mumbo-jumbo about waveforms collapsing at the time of measurement, and the measurement of particle A instantaneously affecting particle B? Why can’t we just assume that spin is simply a static property of each particle from the moment of its creation?

For a simple analogy, suppose I put the Ace of Spades in one envelope and the Queen of Diamonds in another, and send the two envelopes to two different people who live far away from each other. As long as the envelopes are closed, neither of the two people knows what is in their envelope or what is in the other person’s envelope. But as soon as you open one envelope, you can easily deduce what’s in the other envelope as well. Of course, nobody would claim that there is anything mysterious about this, or that the act of opening envelope A caused the card in envelope B to change from a blank piece of cardboard into a particular card. It is obvious that the contents of the envelopes was fixed all the time, even if their recipients did not know which envelope contained which card until they looked inside. So why can’t it be the same with the spin of those quantum particles?

This is called the Hidden Variable Theory. Basically, it says that quantum particles have fixed properties which already exist before you measure them. That doesn’t sound like such a bold assumption, yet apparently the majority of physicists believe it to be false!

The main argument against hidden variables is Bell’s Theorem. To understand what that is about, we need to realize that the ‘spin’ of a particle depends on the axis along which it is measured. If, in the EPR experiment, the recipients of the two particles both measure their spin along the same axis, they will get opposite results 100% of the time. However, if they measure along different axes at a 90 degree angle to each other, there will be no correlation between the results: measuring the spin of particle A will not allow you to predict the outcome of the measurement of B with a success rate of better than 50%. At a minimum, this means that each particle must contain at least two different hidden variables, one for each of the two possible measurement axes.

But it gets worse. A particle does not have just two different possible measurement axes; it can be measured at any angle. For example, if we call the two axes described in the previous paragraph x and y, then either experimenter could just as well decide to turn his spin measurement device to angle z, which is right between x and y, at a 45 degrees angle to each of them. And that’s where things get freaky. Because, if I understand correctly the various explanations I’ve read about this, z is correlated with both x and y in a way that is fundamentally inconsistent with x and y being uncorrelated with each other.

Is it possible that the ‘spin’ hidden variable is not a single value, or a finite collection of values, but a continuous function which describes the spin value that will be measured for any given axis? According to Bell’s Theorem: No, that is not possible. The mathematics get seriously intense here, but basically it seems that, even if we allow every quantum particle to behave as an arbitrarily complex device with lots of internal ‘state’ as we computer guys call it, it is not possible to reproduce the outcome of the actual experiments unless we assume that the particles somehow communicate with each other.