The distance is part of the proof, but not really part of the mystery.
I'm going to throw out an analogy that gets at what's observed and why it's surprising, but doesn't relate to the physics of spin, momentum, position or anything that's actually under observation in these experiments.
It's as if we have a pair of dice, and I throw my die and you throw your die many times. In a classical world, if I throw a three, it has no influence on what you throw; you're equally likely to throw 1-6. But in the quantum world it's as if when I throw a one, your die still has the expected uniform distribution, but when I happen to throw a three, you're a little bit more likely to throw a three. Your die is fair if I happen to roll a one, but it's weighted if I happen to throw a three.
Back in the real world, this is the strange behavior that is observed in experiment. Schroedinger's equation predicts the probabilities perfectly. But Bell shows that it's far from intuitive.
I'm going to throw out an analogy that gets at what's observed and why it's surprising, but doesn't relate to the physics of spin, momentum, position or anything that's actually under observation in these experiments.
It's as if we have a pair of dice, and I throw my die and you throw your die many times. In a classical world, if I throw a three, it has no influence on what you throw; you're equally likely to throw 1-6. But in the quantum world it's as if when I throw a one, your die still has the expected uniform distribution, but when I happen to throw a three, you're a little bit more likely to throw a three. Your die is fair if I happen to roll a one, but it's weighted if I happen to throw a three.
Back in the real world, this is the strange behavior that is observed in experiment. Schroedinger's equation predicts the probabilities perfectly. But Bell shows that it's far from intuitive.