The existence of correlations between events at a distance should not surprise a scientist. The goal of science is to find a rational explanation of correlations. For instance, if I push on a button here, and a light turns on there, I have a causal explanation of the correlation between the two events, invoking an electric circuit that I close and current flowing in the bulb, pushing on the button is the cause of the light shining. Two events can be correlated even if one is not the cause of the other one. For instance, two persons in the same room will see the light shining at the same moment. The perfect correlation between these observations is due to a common cause: somebody pushed on the button and the light was switched on. Can we invoke such a standard scientific explanation to understand the correlations between polarization measurements on EPR correlated photons? At this point, you may ask the question: What is the meaning of "understanding" for a physicist? The answer depends on individuals. Many quantum physicists, following Niels Bohr, would probably answer that a mathematical formalism predicting the observed correlations is enough as an explanation. This is the "shut up and calculate" point of view. But other physicists like to think of explicit mechanisms, which can be represented by images as a complementary explanation. Bill Phillips, a well-known expert on ultra cold atoms, once said: "During the week I belong to the shut up and calculate church, but on Sundays I visit other churches." In fact, even if they pretend to belong to the shut up and calculate school, many physicists use images inspired by the calculation. Think, for instance, of a polarized classical electromagnetic field. Most of us think of arrows in space to represent the vectors, obeying Maxwell equations. Can we similarly build a representation of polarized entangled photons associated with the quantum calculations developed in the previous section? Remember the calculation leading to the probabilities of a joint detection. It consists of projecting the state vector Psi of nu one, nu two, onto the eigenvector plus a plus b, associated with the result of the joint measurement we are interested in. Can we represent that mathematical operation in our space? I doubt it. At least I cannot, since these vectors belong to an abstract Hilbert space, which is a tensor product of the space describing the first photon polarization by the space describing the second photon polarization. A space which does not correspond to ours. So should we renounce building images in the space where we live? It is interesting at this point to recall a strong statement of the late Asher Peres, a well-known theorist of quantum physics. He said, "Quantum phenomena do not occur in a Hilbert space, they occur in a laboratory." I've spent many hours in a laboratory, and I can tell you that it is a place that belongs to our ordinary space. So we should look for an image in this ordinary space where we live. In fact, it is possible to derive an image in our space, based on a detailed quantum calculation. For that, let us assume that the measurement at polarizer one happens first, and then, the other measurement is done at polarizer two. Quantum calculations tell us that at polarizer one we can obtain either plus one or minus one with equal probabilities, whatever the orientation a of the polarizer. To be specific, take this polarizer along x. What happens then after that measurement at polarizer one? You can find the answer in any good textbook on quantum mechanics such as the one of Cohen-Tannoudji and company. You must use the so-called Projection Postulate, which tells you that when a measurement is done, the initial state vector is projected onto the eigenspace associated with the result of the measurement. Let us apply the recipe to our case. Remember that polarizer one is aligned along the x-axis. Just after the measurement yielding plus one for nu one, that is to say associated with x polarization for photon nu one, the initial state vector must be projected on the subspace associated with x one. After normalization, the result is the state vector x_1, x_2. If you do not know how to do that calculation, you will find details in the quiz at the end of this section. The state (x_1,x_2) is factorable, and describes a pair of photons each with a well-defined polarization, photon nu one is polarized along x, which is not a surprise, and photon nu two is also polarized along x. If a measurement is done on nu two with polarizer two along x, one will find plus one. Similarly, if one has found minus one for nu one, that is to say a y polarization, the projection would lead to y_1, y_2, that is to say, a photon nu two polarized along y. A measurement on nu two with polarizer two along x will then yield minus one. We recover the total correlation found with the global calculation for the case of parallel polarizers. If now polarizer two is at a different angle, a straightforward generalization leads to the cosine squared variation found with the global calculation. We have thus found an image in our ordinary space. Should we be satisfied with that image? Einstein was certainly not happy with the description, in which the measurement on photon nu one instantaneously affects the state of photon nu two which, is at a distance. You remember, of course, that Einstein was the discoverer of the relativity theory, in which nothing can travel faster than light, so nothing can be instantaneous. An image such as the one just presented, derived from the standard quantum mechanics formalism, was unacceptable for him. That led him to question the completeness of that formalism. Indeed, one would have a more reasonable image if one would accept to complete the formalism of quantum mechanics with supplementary parameters. Let us suppose, indeed, that the two photons of a pair share from the moment of emission, a common property lambda that will determine the outcomes of the measurements on each photon. The result of the measurement at polarizer one would then be given by a dichotomic function A of lambda a with values either plus one or minus one. Similarly, a function B of lambda b, would give the result of the measurement at polarizer two. Correlations would then be as easy to understand as the similarities of identical twins, who have the same eye colors, sex, diseases, etc. The supplementary properties looking like lambda are obviously the identical chromosomes of the two twins. Coming back to our photons, we can render an account of the randomness of each individual result by introducing a random distribution rho of lambda of the parameters lambda, among the successive pairs. This distribution has the standard properties of a probability distribution: positive and normalized. Some models of this kind can reproduce some strong correlation predicted by quantum calculations for the EPR state. Consider, for instance, a very natural model. The one many of you would invent, I'm sure, after some thinking. Assume that when the two photons are emitted, they have a well-defined linear polarization characterized by the angle lambda with the vertical axis. The result of a measurement at polarizer one, oriented along a, will be given by a dichotomic function A of lambda, a taking the value plus one if the polarization of the photon is at less than Pi over four of the polarizer axis, and minus one if the polarizer is at more than Pi over four. Check that the function written here does the job. A similar function is written for polarizer two. The distribution rho of lambda is taken uniform. A simple calculation which you can find in a document attached to this week lesson, then allows one to calculate the expected probabilities of simple and joint detections. The single detections are found totally random, as predicted by the quantum calculation. The joint probabilities lead to a correlation coefficient plotted here. You see that for angle zero or Pi over two, the simple model yields a full correlation as the quantum prediction. More generally, the quantum predictions are not very different from the ones of the simple model. Such model to understand the EPR correlations is appealing and as rational as invoking chromosomes to understand similarities in identical twins. But this model amounts to completing quantum mechanics since one considers different kinds of pairs with different values of lambda, while in the quantum formalism, all pairs are described by the same state Psi. It turns out that Niels Bohr, who had made immense efforts to develop a consistent interpretation of the formalism of quantum mechanics, was deeply convinced that it was impossible to complete that formalism, and immediately objected the conclusion of the EPR paper. His arguments, however, were not clear enough to convince Einstein. It is fair to say, that the debate between the two old physicists did not attract much attention from active physicists who were using successfully quantum mechanics to interpret their discoveries. In fact, you must realize that it was a debate about interpretation only, not about the mathematical formalism since Einstein fully accepted the results of the quantum calculations. But he thought that these results should be interpreted invoking properties belonging to each photon, in order to avoid the apparent instantaneous influence at a distance at the moment of the measurement. The debate between the two giants lasted until Einstein's death in 1955. Ten years later, the discussion would take a new turn with a far-reaching discovery by John Bell. Einstein's point of view would lead, in some very rare circumstances, to a conflict with quantum predictions.