The essence of quantum entanglement (as exhibited in things like EPR experiments and tests of Bell's inequalities) is that the joint probabilities for combinations of spacelike-separated events depend on the choices of particular measurement operations that are applied to entangled systems at those separate events, and moreover that this dependence is non-linear. It is the non-linearity that makes it impossible to account for the dependence in conventional terms. In order to convey any sense of quantum entanglement, we must first introduce a CHOICE of measurements to be made on each of the two coupled systems, and then we need to examine the correlations resulting from various combinations of choices. For example, suppose we produce a pair of objects in such a way that any one (and only one) of three measurements, A, B, or C, can be performed on its contents. Whichever measurement is performed, the result will be either 0 or 1. Several pairs of objects are prepared in this way, and one member of each pair is sent off to Mars while the other stays here. Then in both locations (here and Mars) we perform one of the three allowable measurements on each object and record the results. Our choices of measurements (A, B, or C) may be arbitrary, by flipping coins or whatever. We find that, regardless of which measurement we decide to make, the chances are 50% of getting "1", and the folks on Mars discover the same thing. This is all that either of us can determine separately. However, when we bring all the results together and compare them in matched pairs, we find the following correlations Earth A B C A 0 3/4 3/4 Mars B 3/4 0 3/4 C 3/4 3/4 0 The numbers in this matrix indicate the fraction of times that the Mars and Earth results agree (both 0 or both 1) when the indicated measurements have been made on the two members of a matched pair of objects. Notice that if Earth and Mars happenned to choose to make the same measurement for a given pair of objects, the results NEVER agree. In other words, they are always the opposite (1 and 0, or 0 and 1). Also notice that the overall probability of agreement if both measurements are selected at random is 1/2. This shows the true nature of quantum entanglement, which is of an entirely different character from classical correlations that may exist between spacelike separated events. For example, classically we could prepare each pair of objects in advance to always give the same results, and this correlation would persist even after the objects have become separated. This is the kind of distant correlations that we deal with constantly in everyday life. However, in the case described above there is no way (classically) of preparing the pairs of objects in advance of the measurements such that they will give the joint probabilities listed above. To see why, notice that each object must be ready to respond to any one of the three measurements, and if it happens to be the same measurement as is selected on its matched partner, then it MUST give the opposite answer. Hence if the Earth object will answer "0" for measurement A, then the Mars object MUST answer "1" for measurement A. Likewise for the other measurements, so there are only eight ways of preparing a pair of envelopes Earth Mars A B C A B C a 0 0 0 1 1 1 b 0 0 1 1 1 0 c 0 1 0 1 0 1 d 0 1 1 1 0 0 e 1 0 0 0 1 1 f 1 0 1 0 1 0 g 1 1 0 0 0 1 h 1 1 1 0 0 0 These preparations, and ONLY these, will yield the perfect anti- correlation when the same measurement is applied to both objects. Unfortunately, if we simply randomly select one of these eight preparations (with equal probabilities) for each pair of objects, we won't match the other correlations predicted by quantum mechanics. Instead we get Earth A B C A 0 1/2 1/2 Mars B 1/2 0 1/2 C 1/2 1/2 0 Notice that the overall probability of agreement if the two measurements are selected randomly is NOT 1/2 (as quantum mechanics says it must be). Instead, it is 1/3, so this isn't right. Still we might imagine that some other selection strategy for choosing from the eight possible preparations might give the right overall results. However, regardless of our strategy, the overall preparation process must result in some linear convex combination of the eight cases, i.e., there must be positive constants a,b,..,h whose sum is 1 and such that the "agreement probabilities" in the first table are satisfied. This implies, among other things, that c+d+e+f = 3/4 b+d+e+g = 3/4 b+c+f+g = 3/4 Adding these up gives 2(b+c+d+e+f+g) = 9/4, and so the sum of the coefficients b through g is 9/8, which exceeds 1. Hence there is NO linear combination of those eight preparations that can yield the joint correlations predicted by quantum mechanics. This is a typical "Bell inequality", and its violation illustrates that the questions presented by the phenomena of quantum mechanics are deeper and more profound than is sometimes realized. Nevertheless, it remains true that no superluminal transfer of information is implied. This is actually one of the most interesting features of this analysis, because it shows that in some circumstances our classical understanding demands that superluminal communication "must have occurred" in order to give the observed results, even though in fact there is no *effective* transfer of information at all. See Quantum Entanglement and Bell's Theorem for a more detailed discussion of these issues.

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