Although it was not the example chosen by Einstein Podolsky and Rosen in the 1935 paper where they pointed out entanglement and its extraordinary properties, pairs of entangled spin one-half, or of polarization entangled photons, are privileged systems to discuss entanglement. The reason is that the formalism and the calculations are simple in the two-dimensional state-space of each particle and nevertheless allow one to grasp the subtleties of entanglement. Spins one-half and polarized photons are formally equivalent and since we are in quantum optics course, I have chosen the polarized photon case. There is another important reason for that choice. Polarized photons have been used in many landmark experiments about entanglement and are still key components of many quantum information protocols, while experiments with spin one-half particles are exceedingly difficult. The figure shows a source S emitting two photons nu one and nu two propagating in opposite directions minus z or plus z. The two photons have different frequencies, so there is no question of indistinguishability here. Polarizers one and two followed by detectors not indicated on the figure, allow one to measure polarizations of each photon along orientations indicated by the unit vectors a and b at angles alpha and beta from the x-axis respectively. We assume that the polarization state of the pair is described by the state vector x_1, x_2 plus y_1, y_2 with a normalization factor, one over root two. The ket x_1, x_2 represents a pair of photons propagating to minus z and plus z with the same polarization x. Obviously, the ket y_1, y_2 represents a pair of photons propagating to minus z and plus z with the same polarization y. We will refer to the total state as an EPR state for Einstein Podolsky Rosen. What is special about that state? It is a superposition of two valid two-photon states and it is therefore a valid two-photon state. But in contrast to the states x_1, x_2 or y_1, y_2, one cannot attribute a specific polarization to the photon one going to the left and a specific polarization to the photon two going to the right. As a matter of fact, if it was the case, it would be possible to factorize the state psi, that is to say, write it as a tensor product of a polarization for photon one and a polarization for photon two. Let me show you that it is not possible. In the expression here, we have used the most general form of polarization for each photon 1 and 2. Developing that expression, we obtain the decomposition of the product state on the complete basis of the state space of the two photons polarizations which is composed of the four kets. (x_1, x_2), (x_1, y_2), (y_1, x_2), (y_1, y_2). Can this product state represent the state psi? Let us try to make all of the four coefficients equal to the coefficients of the EPR state. In psi_EPR, there is neither term (x_1,y_2) nor term (y_1, x_2). So the products gamma nu and lambda mu must be null. But the terms x_1, x_2 and y_1, y_2 must both exist in the expansion of psi, so the product gamma mu lambda nu must be different from zero, which is contradictory. So there is no decomposition of that type possible and thus no possibility of factorization. What we have just seen corresponds to the definition of an entangled state of two particles. An entangled state cannot be expressed as the tensor product of a state of the first particle by a state of the second particle. Beyond mathematics, it means that I cannot wave my hands saying "These photons has a polarization like this and that photon has a polarization like that." But what can I say then about that state? In quantum mechanics, when you know the state of a system, you can calculate the probabilities of the outcomes of any measurement associated with a measurable observable. Here, we can measure the polarization of each photon. Consider then the first polarizer aligned along a. You can obtain either +1 corresponding to a polarization epsilon at an angle alpha or -1 corresponding to the orthogonal polarization. Similarly, for polarizer two, you can obtain either +1 associated with the polarization along b or -1 associated with the orthogonal polarization. At this point, we could calculate the probabilities of obtaining plus one or minus one at one polarizer or at the other one. That is to say, single detection probabilities. It turns out that it is simpler to start with joint measurements which are complete measurements. So we ask for instance, "For a pair described by psi EPR, what is the probability to obtain +1 for photon 1 and +1 for photon 2 with polarizers along a and b respectively?" We note that probability P plus plus of AB. Just apply the basic rule of standard quantum mechanics: write the eigenstate associated with the measurement, project the state on it, and take the squared modulus. I let you do the calculation using the expressions of plus a and plus b. In principle, you should write explicitly the indices one and two to indicate which photon is concerned. But if you follow the rule to write photon one quantities first and photon two quantities second, you can dispense yourself of writing the indices one and two. If you noticed from the beginning that the only useful terms are xx and yy, the calculation is quite simple, and using elementary trigonometry, you find P++(a,b) equal to one-half cosine squared of alpha minus beta. A similar calculation shows that p minus minus is equal to p plus plus. I recommend that you also do the calculation for p plus minus and p minus plus. Using these results, we will now show the most interesting feature of entangled states: strong correlations in measurements. What can we say about polarization measurements on one photon only. Once again, go to your favorite quantum mechanics course. Since it is a partial measurement, you must add the probabilities of all complete measurements associated with a partial measurement you are interested in. To be specific, let us consider polarization measurements on photon nu one. The probability to obtain plus one for polarizer one aligned along a is the sum of P++(a,b) and P+-(a,b). Using the joint measurement probabilities just calculated, you obtain one-half . Similarly, the probability to obtain minus 1 is the sum of P-+ and P--, which is also one-half . It means that whatever the orientation a of Polarizer I, the probabilities of finding a polarization along a or perpendicular to a are equal. Photon nu one appears to be fully unpolarized. Similar results are obtained for photon nu two, which have equal probabilities to be found polarized along b or perpendicular to b, whatever the orientation b. We have thus found that each individual polarization measurement returns a random result, either plus 1 or minus 1. But, consider the expressions of join measurements in the case of parallel polarizers, that is to say, the angle (a,b) equals zero. The values for P plus plus is one-half . Similarly, P minus minus is one-half . I claim that this means a total correlation of the measurements on nu 1 and nu 2. Can you tell why? One can give several good reasons to conclude that the random results at one and two are totally correlated. Consider, for instance, the fact that the signal probability to obtain plus 1 on nu 1 is one-half . The joint probability to have plus 1 on nu 1 and plus 1 on nu 2 is also one-half . It means, that the conditional probability to find plus 1 for nu 2 if 1 finds plus 1 for nu 1 is 100%. If you are not yet fully convinced, consider the joint probability P plus minus, which is zero. It means that if you find plus for nu 1, you are sure not to find minus for nu 2. In other words, you can find randomly plus 1 or minus 1 for nu 1 and similarly for nu 2. But if you find plus 1 for nu 1, then you'll find plus for nu 2, and if you find minus for nu 1 you find minus for nu 2. The random results are fully correlated. Are you surprised? Physics is a quantitative science, and in order to go further, we must express quantitatively the degree of correlation between the results of measurements at polarizers I and II for any set of angles. We will use the classical definition of the correlation coefficient for two random variables with values plus or minus 1 at one and two. To make things concrete, you may think of operators flipping a coin and associating plus one with the head and minus one with tails. The correlation coefficient is the average of the product minus the product of the averages, normalized. In this expression, the overbar is a classical, statistical, ensemble average. If the two random variables are independent, uncorrelated, the average of the product is equal to the product of the averages, and the correlation coefficient is expected to be zero, as should be the case for to honest players using fair coins. But, if the two random outcomes are not independent, the average of the product is different from the product of the averages, and the correlation coefficient will be different from zero. For random variables with equal probabilities to obtain plus 1 or minus 1, the average values are null, and the averages of the squares are equal to one. The correlation coefficient is then nothing else than the average of the product A.B . Considering the four possible values for the product A.B you can express it as a function of the four joint detection probabilities, P plus plus, P minus minus, P minus plus, and P plus minus. We will use this expression to calculate the EPR pairs correlation predicted by quantum mechanics. For EPR correlated photons and parallel polarizers, the quantum calculations gives P plus plus equals P minus minus equals one-half, and P plus minus equals P minus plus equals 0. The correlation coefficient is then one, meaning a total correlation. But what do we have for an angle different from zero? You have calculated the probabilities of joint detection for polarizers at any angles. Using the general expression of the correlation coefficient, you can calculate the correlation expected for EPR pairs. After one line of trigonometry, you will find cosine of twice the angle between polarizers. This is the representation of the expected coefficient of correlation of polarization for the EPR state. The correlation is total when the modulus of the coefficient is one. That is to say, for the polarizers parallel or at Pi over two. In the latter case, the minus one coefficient reflects the fact that when one has plus one on one side the result is minus one on the other side. In fact, it corresponds to the same polarization, since the polarizers are crossed. We now have the full variation of the correlation coefficient, and we confirm that there are angles where the correlation is total. How can we understand these correlations?