Let's say particles have a 'direction angle' that we can measure with a detector that only gives 'up' or 'down' relative to a direction angle measurement. We ca

Let's say particles have a 'direction angle' that we can measure with a detector that only gives 'up' or 'down' relative to a direction angle measurement. We can change this direction angle measurement with a knob to set what the measured 'up' and 'down' answers are relative to the detector's direction angle. Further let's say particles can be quantum entangled so that when when two detectors are placed very far apart, many light years apart, say, and measure a quantum entangled pair of particles. When the two detectors are set to the same, but arbitrary, angle, the detectors give the same answer. This is normal correlation. Quantum correlation says that as one dial moves away from the other reference point, the correlation falls off as a sine wave, not a linear decrease as would be expected by classic probability. To see how bonkers this is, do the following experiment: Set detector X to be at angle 0 and detector Y to give a 1% error rate. Call that 1% angle 'a'. So a sample experiment run might be: X(0): 0001000101001110...111010 Y(a): 0011000101001110...111010 In the above, 1 could be an 'up' and 0 could be a 'down' detection, say. For concreteness, let's just say A and B ran 100 detections and there was one difference between them (giving 1% error), represented by the third differing bit in the above. Now let let's change both X and Y by the same angle so the relative error rate between them is still 1%, this might give something like: X(a): 0011000001001110...111010 Y(2a): 0011000101001110...111010 X and Y still have one difference in the above, but now with the 8th position changed. So far this is nothing unexpected from classical probability. Now, we know that from X(0) to Y(a) there's one change, from X(a) to Y(2a) there's one change. Classic probability says that there can be at most two flipped bits from X(0) to Y(2a). Quantum mechanics predicts three. To convince yourself, try making a list of bits such that there's one difference between X(0) and Y(a), one difference between X(a) and Y(2a) but three differences from X(0) to Y(2a). It's impossible and this is the heart of Bell's theorem. Bell's theorem is a classical probability statement, generalized from my above statement that if |X(0)-Y(a)|=1, |X(a)-Y(2a)|=1 then |X(0)-Y(2a)|<=2. Quantum entanglement violates Bell's inequality. The 0 reference point has to be arbitrary (in the above it should really be X(ref_angle + a), Y(ref_angle + 2a), etc.) and you have to assume no faster than light communication (that is, independence) to get the contradiction. There are some further subtleties with the above argument but hopefully that's intuitive enough to follow why quantum entanglement is so counter intuitive. EDIT: corrected X(a) bit string