Quantum Entanglement, Dark Counts, Coincidence Detection

[caraher]

45. OK, I get it now

At least the original version of the experiment. It's very interesting and well worth doing, but I think Zeilinger's explanation is spot-on, and considering all the variations is a great exercise in thinking about causality.

First, I now understand why Cramer is looking at momentum - it's because the experiment needs to use two conjugate variables, of which one pair is position and momentum. When you get the interference pattern you're measuring momentum eigenstates, and when you get the 2 peaks (in the Dopfer experiment) you're measuring position eigenstates. To make this all work you always have to measure two things governed by an uncertainty relation.

Next comes the question of coincidence measurement. At first I was thinking about this experiment as involving spacelike separation of the measurement events, but I see that in order to make a time travel connection that's not the case - you need the measurement you are "choosing" to have timelike separation from the other measurement, otherwise all you're doing is repeating the coincidence measurement version and reconfirming the nonlocality of QM but without showing any backwards-in-time causality. Cramer's argument is very seductive - my initial reaction was that adding 10 km of fiber to the optical path shouldn't change the result. And if that's so, it's undeniable that the choice of which measurement to make 50 microseconds after the fact does indeed determine the measurement made first!

I also had some fuzzy notion that the need to sample many points to prove you have an interference pattern was somehow crucial, but by mentally extending that 10 km of fiber to an astronomical distance I convinced myself that this does not matter - you could have enough photons in transit to the distant collector that their partners could all be fully sampled and the interference pattern collected well before the first photon in the stream arrived at the distant detector.

I think where the Cramer experiment falls apart lies precisely in a genuine need to collect photons in coincidence. The issue is that we don't get to tell nature not to collapse the entangled wavefunction when it interacts with the near detector. In the Dopfer experiment, the near detector sits behind a double-slit, with the result that if you do not record only photons in coincidence you will always get a 2-slit pattern at the near detector. ("Likewise, registration of photon 2 behind its double slit destroys any path information it may carry and thus, by symmetry, a Fraunhofer double-slit pattern is obtained for the distribution of photon 1 in the focal plane behind its lens, even though that photon never passed a double slit (Fig. 4)!&quot You do, of course, get to choose which measurement you make at the far detector. If you choose to make the momentum eigenstate measurement, you will also get an interference pattern, because measuring the momentum eigenstate at the near detector (which is what's happened already at the near detector) means you wind up with a momentum eigenstate at the far detector.

If you choose to make the position eigenstate measurement at the far detector, you won't get two peaks - you'll get a smear that reflects the fact that you're measuring the momentum eigenstate that now describes the far photon. The welcher-weg information was erased when the initial measurement occurred at the near detector.

One could reverse the setup, making the path with the double slit longer. Then what happens is unsurprising - when you measure with the Heisenberg microscope set up with the detector at a distance f from the lens, you get an interference pattern; if you set it up at 2 f, you get welcher-weg information but no interference.