I have to say that the sequence on Quantum Mechanics has been awfully helpful so far, especially the stuff on entanglement and decoherence. Bell’s Theorem makes a lot more sense now.
Perhaps one helpful way to get around the counterintuitive implications of entanglement would be to say that when one of the experimenters “measures the polarisation of photon A”, they’re really measuring the polarisation of both A and B? Because A and B are completely entangled, with polarisations that must be opposite no matter what, there’s no such thing as “measuring A but not measuring B”. A and B may be “distinct particles” (if distinct particles actually existed), but for quantum mechanical purposes, they’re one thing. Using a horizontal-vertical basis, the system exists in a combination of four states: “A horizontal, B horizontal”, “A horizontal, B vertical”, “A vertical, B horizontal”, “A vertical, B vertical”. But because of the physical process that created the photons, the first and fourth components of the state have amplitude zero. On a quantum level, “measuring the polarisation of A” and “measuring the polarisation of B” mean exactly the same thing—you’re measuring the state of the entangled system. The two experimenters always get the same result because they’re doing the same experiment twice.
(Of course, when I say “measure the thing”, I mean “entangle your own state with the state of the thing”.)
After all, most practical experiments involve measuring something other than the actual thing you want to measure. A police radar gun doesn’t actually measure the speed of the target car, it measures the frequency of a bunch of microwave photons that come back from the target. Nobody (especially not a policeman) would argue that you aren’t “really” measuring the car’s speed. Imagining for a moment that the car had any kind of macroscopic spread in its velocity amplitude distribution, the photons’ frequency would then be entangled with the car’s velocity, in such a way that only certain states, the ones where the car’s velocity and the photons’ frequency are correlated according to the Doppler effect, have any real amplitude. Thus, measuring the photons’ frequency is exactly the same thing as measuring the car’s velocity, because you’re working with entangled states.
If, on the other hand, the pair of photons were produced by a process that doesn’t compel opposite polarisations (maybe they produce a pair of neutrinos, or impart some spin to a nearby nucleus), then the four states mentioned above (A-hor B-hor, A-hor B-vert, A-vert B-hor, A-vert B-vert) all have nonzero amplitude. In this situation, measuring the polarisation of A is not an experiment that tells you the state of the system—only measuring both photons will do that.
I have to say that the sequence on Quantum Mechanics has been awfully helpful so far, especially the stuff on entanglement and decoherence. Bell’s Theorem makes a lot more sense now.
Perhaps one helpful way to get around the counterintuitive implications of entanglement would be to say that when one of the experimenters “measures the polarisation of photon A”, they’re really measuring the polarisation of both A and B? Because A and B are completely entangled, with polarisations that must be opposite no matter what, there’s no such thing as “measuring A but not measuring B”. A and B may be “distinct particles” (if distinct particles actually existed), but for quantum mechanical purposes, they’re one thing. Using a horizontal-vertical basis, the system exists in a combination of four states: “A horizontal, B horizontal”, “A horizontal, B vertical”, “A vertical, B horizontal”, “A vertical, B vertical”. But because of the physical process that created the photons, the first and fourth components of the state have amplitude zero. On a quantum level, “measuring the polarisation of A” and “measuring the polarisation of B” mean exactly the same thing—you’re measuring the state of the entangled system. The two experimenters always get the same result because they’re doing the same experiment twice.
(Of course, when I say “measure the thing”, I mean “entangle your own state with the state of the thing”.)
After all, most practical experiments involve measuring something other than the actual thing you want to measure. A police radar gun doesn’t actually measure the speed of the target car, it measures the frequency of a bunch of microwave photons that come back from the target. Nobody (especially not a policeman) would argue that you aren’t “really” measuring the car’s speed. Imagining for a moment that the car had any kind of macroscopic spread in its velocity amplitude distribution, the photons’ frequency would then be entangled with the car’s velocity, in such a way that only certain states, the ones where the car’s velocity and the photons’ frequency are correlated according to the Doppler effect, have any real amplitude. Thus, measuring the photons’ frequency is exactly the same thing as measuring the car’s velocity, because you’re working with entangled states.
If, on the other hand, the pair of photons were produced by a process that doesn’t compel opposite polarisations (maybe they produce a pair of neutrinos, or impart some spin to a nearby nucleus), then the four states mentioned above (A-hor B-hor, A-hor B-vert, A-vert B-hor, A-vert B-vert) all have nonzero amplitude. In this situation, measuring the polarisation of A is not an experiment that tells you the state of the system—only measuring both photons will do that.