Bell's Inequality

Einstein's EPR experiment, and Bell's attempt to clarify the significance of its results, have often been misunderstood and badly reported. This essay attempts to separate fact from fiction, and without sacrificing rigour, to present the concepts in a way that is easily understood.



When Albert Einstein, Boris Podolsky and Nathan Rosen devised their well-known "EPR thought experiment" in 1935 (see [1]), they could not have guessed that sixty years later it would still be a topic for profound disagreement, or that generations of physicists would have failed to understand its real significance. Experimental technique in 1935 was inadequate to permit the actual performance of the experiment, but today more sophisticated procedures are commonplace, and in recent years many comparable tests have been carried out.The details have usually differed from those specified in the original EPR paper, but only for convenience. There is now little doubt what results to expect whenever this sort of experiment is performed. Although these results are highly perplexing, they are unambiguous. But over the years there has been widespread misunderstanding of their true significance; to my knowledge only one writer has correctly assessed the conclusion we should draw from them.

Much of the experimental basis for quantum theory was understood in 1935, including the "uncertainty principle". It was known, for example, that we cannot measure accurately both the position and the momentum of an elementary particle; whatever methods we adopt, the more accurately we measure one, the less precise must be the value we can obtain for the other. Nature seems to ration strictly the information it makes available from such a particle. Einstein knew, however, that we can produce pairs of identical particles travelling in opposite directions, which must necessarily have the same speed. Can we not measure the position of one and the momentum of the other, and thus defeat nature's restrictions? Quantum theory predicted we could not, for a measurement applied to one particle must necessarily affect the value of the attribute we wish to measure for the other one. Einstein and his colleagues realised this implied that the two particles must, in some way, cooperate to conceal information from the experimenter, even when they are separated by so great a distance that no signal should be able to pass between them. The measurement we make on one of the particles must instantaneously affect the behaviour of the other. This sort of cooperation, or "ghostly action at a distance" as Einstein called it, seems quite impossible, particularly as it might sometimes have to act at speeds greater than that of light, which Special Relativity theory had shown could never occur. This proves, they argued, that Quantum Theory is wrong.


Some of the EPR experiments which have been performed recently have involved light photons and polarising filters. Such experiments are easier to perform, and the results perhaps easier to comprehend, than those using position and momentum. A polarising filter has an "axis of polarisation" whose orientation we can know. And each photon can be thought of as having a "spin axis", whose direction we generally cannot know, and so must regard as being random. When a beam of random photons meets a polariser, it is found that about one half pass through, and the remainder are stopped. But of those that do pass through, their later behaviour seems to indicate that as they are transmitted their spin-axes all get turned to be parallel to the axis of the polariser. If these photons then strike a second polariser with its axis parallel to the first, all of them will be transmitted. If the second has its axis perpendicular to the first, then none are transmitted. In general, if the axes of the two polarisers are inclined at an angle theta to each other, the proportion of photons transmitted by the first polariser which pass also through the second is found by squaring the value of cos(theta).

When this process is used as the basis for an EPR experiment, atoms of calcium are used to generate the photons. These atoms can be put into an unstable energy state such that, when this decays, a pair of identical photons are emitted in opposite directions, just as we require. We do not know the direction of the spin-axes of these photons, but we do know they must be identical. We set up two polarisers, one in the path of each photon, and each able to have its axis set in any direction we choose. Modern technology allows us to track the behaviour of individual photons, and to determine the fate of each pair as they traverse the apparatus. The effect we observe is fully in accord with quantum theory, and opposed to Einstein's view, and does seem very strange. As one of the photons passes through a polariser, not only is its own spin axis changed, but the other photon is found to behave as if it too had encountered this polariser, however far away it might then be. Each photon appears to send to its twin a message describing the orientation of the polariser it is encountering, and its own reaction to it. The orientation of each polariser thus affects the probability that the other photon will pass through its polariser. If the axes of the two are parallel, then the photons are either both transmitted or both absorbed. If they are perpendicular, then one is transmitted and the other absorbed. And for any other angle, the probability that both are transmitted or both absorbed is again found by squaring the cosine of the angle between the two axes.

Clearly the behaviour of the two photons is related in some way, but here is the crucial question; could this degree of cooperation be due to a pact the photons had agreed upon before separating, or does it show there must be some communication between them subsequently?


The manufacturers of Gofast automobiles use an unusual method of selling their cars to dealers. Every Saturday morning they hold auctions at various centres around the country, where dealers can bid up the price they will pay for the limited number of cars available that day.

There are two partners in the Squaredeal Motors business. Every Saturday they set off in opposite directions to different auctions, from which they each may, or may not, return with a car, depending upon the demand and the price reached in the bidding.

Joan, the secretary of Squaredeal Motors, noticed a surprising degree of agreement between the two partners in the decisions they make. Sometimes both will purchase cars, sometimes only one will, and sometime neither. But if the auction price at the two locations happen to agree, either they both bring back a car, or neither does. Furthermore, if the prices at the two auctions differ, one partner never buys a car at a higher price than the other partner has decided is too high for that week. Joan knew the partners did not carry mobile phones, and concluded that they must be able to communicate by telepathy.

Then one day they told her their simple secret. Each Saturday, before going their separate ways, they agree on the maximum price they will pay that day. No further communication is needed; they do not have any supernatural powers.


Now could our two photons achieve their performance just be agreeing tactics before separating, or must they engage in "telepathy"?

A possible tactic might be for the photons to agree their (common) spin-axis before setting off. They could also agree that, if they encounter a polariser whose axis makes an angle of less than 45 degrees with their own axis, they will pass through, but if more than 45 degrees they will not. It is easy to see that photons with such a policy will have a 50% probability of getting through a polariser, as we know they should. And if the two photons encountered polarisers with their axes parallel, we can easily see that they would either both be transmitted or both stopped. It is only a little more difficult to see that, with polarisers whose axes are perpendicular, one would be transmitted and one stopped.

What happens if the angle between the polariser axes is 45 degrees? It is not too difficult to work out that they will respond in the same way as each other, both transmitting or both stopping their respective photons, with a probability of 50%. And this again is what quantum theory predicts, for the square of the cosine of 45 degrees is 1/2. It is beginning to look as if EPR does not require us to believe in telepathy.

But wait! Consider two polarisers at 30 degrees. Since the square of the cosine of 30 degrees is 3/4, we would expect 3/4 of our photon pairs to agree, and 1/4 to disagree. But careful consideration of the pact they agreed upon, shows that agreement would occur in 2/3 of cases, not 3/4. They have got it wrong; their little plan is not clever enough. They would still need telepathy to obey the quantum rules. It is not easy to discover whether there is any other sort of pact which would work in all cases, but by 1964 nobody had managed to devise one.


It was then that John S. Bell published a remarkable mathematical equation (or rather an inequation) (see [2]), very general in nature, by which we can test any experiment of this type for "telepathy". Bell's inequality can take many different forms, but the following is one of the simplest.

Suppose there are two measurements we can make at the left hand end, which we shall call W and X (each with a "yes/no" answer, such as the transmission of a photon), and two at the right hand, called Y and Z. We define a type of correlation coefficient, q, by means of a simple formula, of which the following is an example:

q(XY) = prob(X and Y are same) - prob(X and Y are different)

It will be seen that q values must lie between -1 and +1, just like other more conventionally defined correlation coefficients. Then Bell's result can be written as follows:

|q(YW)-q(ZW)| + |q(YX)+q(ZX)| <= 2

Applying this to our present problem, with twin photons and two polarising filters, the patient reader might like to show that this formula is satisfied with certain combinations of polariser axes, and infringed with others. But the following argument is perhaps simpler to follow:

Suppose the polariser on the left can take any of the positions A, B, C or D, these being at 30 degrees to each other as shown, and those on the right, A', B', C' and D', are respectively parallel to them. For our first experimental run, set the left detector to direction A and the right one to B'. The angle between them is 30 degrees and so, as shown above, the probability of agreement is 3/4; with a sufficiently long series of readings, about 3/4 of the results at A, whether "yes" or "no", will be the same as those at B'. Only 1/4 will be different, which means that the list of A results would need only 1/4 of them changing to give the B' results.

Now suppose that instead of setting the left polariser to A, it had been at C. If the right hand polariser remained at B' the run of results there would be the same as before, because we are assuming the position of the left hand polariser cannot influence the results at the right hand (without telepathy). The angle between the detectors is still 30 degrees, so again only 1/4 of the right results will differ from the left.

Thirdly, turn the right polariser to D', leaving the left one at C. Once again only 1/4 of the D' results will differ from the C results.

To summarise, only 1/4 of the A results differ from the B' results, only 1/4 of the B' results differ from the C results, and only 1/4 of the C results differ from those at D'. It follows that no more than 3/4 of the A results can differ from those at D'. But this is wrong; D' is perpendicular to A, and so all its results should differ. This contradiction is, in effect, an example of the sort displayed by infringements of Bell's Inequality.

What exactly is the significance of this infringement? The usual analysis goes something like this: Bell made two assumptions in deriving his inequality. The first of these, which we call "locality", is that a measurement we make at one place cannot affect the result we obtain at another if they are so far apart that speed-of-light signalling is impossible. The second is often called "reality", that the things we measure in quantum experiments, such as the direction of a photon's spin or the momentum of a particle, actually have real values even when we are unable to find them precisely. Strong evidence in favour of this is provided by the fact that in some circumstances we can actually measure their values exactly. For example, we can find a momentum as accurately as we wish if no positional information is required at the same time, so it appears that momentum is a precise characteristic of a particle. Likewise we know accurately the spin axis of a photon after it has passed through a polariser.

So the fact that Bell's Inequality is broken implies that one or both of these assumptions, locality or reality, must be false. It is because the abandonment of either of these "obvious" precepts is so strongly opposed to intuition, and involves such a major change in one's philosophy, that so much effort has been devoted to testing the EPR results experimentally, and so much ink and sweat devoted to their analysis.

Such was the standard response to the EPR experiments, and to Bell's interpretation of them, until Thomas Brody exposed a serious flaw in the argument.


In his book "The Philosophy behind Physics" (see [3]). Brody shows conclusively that, although Bell did make the above two assumptions in deriving his formula, he did not need to do so. The inequality can be derived from a purely logical argument, making no reference to any experiment. The results of the EPR experiments remain perplexing, but the perplexity stems not from considerations of locality or reality. Brody points out that the inequality is broken only when we make several runs of measurements with different settings of our apparatus, such as different orientations of the polarisers described above, and then assume that the data can be assembled into a single probability distribution. It is this "Joint Measurability Assumption" which is flawed.

In the experiment illustrated by the diagram above we took a run of observations with the polarisers set at A and B', and then a second run at B' and C. We wrote, "If the right hand polariser remained at B', the run of results there would be the same as before ..." This was our mistake, a mistake which allows us to prove the impossibility of a phenomenon which is observed regularly now in our laboratories. There is no possible way of verifying the assertion that the results would have been the same, and I try to show below that from more than one viewpoint it must be regarded as meaningless.


Brody discusses in detail the validity of a "Joint Measurability Assumption", in his learned and thorough treatise. But there are several simpler ways of exposing the fallacy in our argument.

In another essay on the present website,


it is stressed that anomalies often result from probability arguments when the sample space of outcomes is not defined clearly, and an amusing illustration was provided in that essay. The present problem can be regarded as a further (and more serious) illustration of this principle. In our first run of results, with the polarisers set at A and B', the sample space for our probabilities was the set of four possible outcomes; the photons at A and B' were each either transmitted or absorbed. Then for the second run we had a new sample space, relating to B' and C. There is no way in which these can be combined into a single sample space, for the quantum rules forbid the "measuring" of the spin of any particular photon in both the A direction and the C direction. We can obtain only "one bit" of information concerning the spin of a photon, and our tacit assumption of "joint measurability" contravenes this restriction.

I provide an alternative analysis in another essay on the present website, entitled

Conditionals and Counterfactuals

Here it is pointed out that some examples of counterfactual statements are essentially meaningless. No better example can be provided than that quoted above. "If the right hand polariser remained at B' ..." is the start of a totally void counterfactual.

The behaviour of identical pairs of photons is certainly strange, but the usual analysis fails to point out the essence of this strangeness. The EPR experiment tells us nothing about the "reality" of quantum quantities, or, what is sometimes assumed to be the same thing, about their possible "hidden variables". Nor does it provide an example of faster-than-light transfer of information which we could use for exciting "science fiction" adventures. There is a sense in which the two photons remain a single entity despite their distance apart, and what happens to one is correlated with what happens to the other, in a way that we may still wish to call "non-locality". The influence is not directly on the behaviour of the photons, but only on the probabilities associated with their behaviour. It can carry no information. This phenomenon is certainly counter-intuitive, but it need not have so devastating an effect on our philosophy and beliefs as some writers have suggested.

A much wider-ranging discussion of the quantum paradoxes will be found on another essay in this series, entitled

Time and Quantum Measurement

In this, an approach is developed which it is hoped will make some of the ideas put forward on these pages more plausible. Two of the questions which may have caused particular difficulty, the sudden change of spin-axis of a photon on passing through a polariser, and the idea of non-locality, are discussed in more detail there. I believe the problems of interpreting quantum theory, although certainly not trivial, pose less threat to our understanding than has been implied by some of the more sensational writers on the subject. But these phenomena are the most amazing and fascinating that man has ever encountered; in coming to terms with them we must not try to explain them in terms of our everyday experiences.

(c) Hector C. Parr (1997)


1. A. Einstein, Physical Review 47, pp.777-780 (1935)

2. J. S. Bell, Physics, Vol.1, p.195 (1964)

3. T. Brody, The Philosophy behind Physics, (Springer-Verlag 1993)

There are many websites devoted to the EPR experiments and Bell's inequality. I select a few only, not because I necessarily endorse their interpretation, but because they are very clearly written.

The Alternative View, by John G. Cramer
The EPR Paradox, by John Blanton
An Effective Test of Bell's Inequality
EPR Paradox - Bell's Inequality, by David Elm
Three Experimental Tests ... , by Alain Aspect


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