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I have been watching Feynman's fantastic lecture on the double slit experiment with electrons:
https://www.youtube.com/watch?v=kEx-gRfuhhk&t=15046s

Feynman proposes to shine light on the slits to detect which slit the electrons have travelled through and explains that the interference pattern would disappear - see here.

My question is, if we erase the which-way information will the interference pattern appear?

In particular - Feynman calls the detection "a flash of light" - suppose we recombine the paths taken by the "flash of light" photons (e.g. with prisms) into a single path such that it is impossible to tell from which slit they originated, will the interference pattern show up?

My guess is that it should be impossible to do an erasure that will restore the electron's interference pattern, since if it was possible then it would also be possible to set up a version of the delayed erasure experiment with retro-causality that cannot be explained away - https://en.wikipedia.org/wiki/Delayed-choice_quantum_eraser#Consensus:_no_retrocausality

So, what is the answer?

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  • $\begingroup$ Be careful with the eraser experiment, in general it has been debunked, there is no faster than light and no retro-causality. $\endgroup$ Commented May 16 at 21:19
  • $\begingroup$ In your expt you would not need to recombine the photon beams, equivalently you could just not observe/detect the photon beams. But by your expt definition an electron has interacted with a photon, all that means is the electron got interrupted on its way to make the "interference" pattern and now just generates a new random pattern on the screen. $\endgroup$ Commented May 16 at 21:23

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Let's ignore your prism and instead use just one detector that sees(detects) all the electrons either from slit 1 or 2 .... in this case the electrons will form the pattern .... and this is similar to your prism setup. In both cases we have no which way information .... the electrons are free to choose either path .... the forces (virtual) creating the paths are effected by both slits which results in the pattern.

What is fundamental here is that with 2 detectors the electrons must form paths (or wave functions) with a unique detector and photon .... by adding unique detectors the electrons have no choice but to make wave functions unique to each slit .... the electrons and virtual forces only see/choose one slit or the other.

Another way of stating this is that the dynamic EM field which forms the path the electron must take makes a solution that involves a unique detector. When the detection is not unique the EM field is influenced by both slits in which the wave function solutions are the "pattern". We can even call this the Feynman pattern as he explains it with the path integral theory.

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  • $\begingroup$ so are you saying that with a "flash" detector that can't tell you which way, the interference pattern will be generated? $\endgroup$
    – nir
    Commented Jul 18, 2022 at 5:49
  • $\begingroup$ Yes, there is a historical but incorrect belief that nothing happens before electron/photon emission and we can't explain why they take certain paths. But nothing could be further from the truth, the excited electron before emission or photon generation can be in the excited state for quite a while and all the while generate forces in the EM field .... thereby influencing and determining its path. The EM field is very dynamic and everything is happening at the speed of light. If we consider this fact there are no mysteries with particles making "patterns" ... or eraser experiments and such. $\endgroup$ Commented Jul 18, 2022 at 12:43
  • $\begingroup$ But what if the light from the "flashes of light" travels a long distance before recombining? So the electron hits the screen after a few nanoseconds, but the wave function for the photon doesn't hit its detector for several minutes? $\endgroup$ Commented Sep 6, 2022 at 14:33
  • $\begingroup$ The whole theory also requires that the photon also be detected by (Ccd, PMT) to cancel the pattern. If we move the CCD far away the probability of a CCD interaction would fall ... but a small amount could still be detected. The main point is that the EM field is already involved, even before electron emission, by the excited electron in the electrode .... The EM field is guiding everything. When the detector is far and we have the luck to get a detection it just means the electron emission was more delayed until the EM field virtually had the sensed the forces of the CCD, photon, electron. $\endgroup$ Commented Sep 6, 2022 at 21:46
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This is a case where Feynman's framework gives a different answer than what the traditional quantum mechanics framework gives. Feynman says:

What we will call “an event” is, in general, just a specific set of initial and final conditions. (For example: “an electron leaves the gun, arrives at the detector, and nothing else happens.”)

and

When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference

In your example, there's a detector on the electron screen, and a separate detector in the single path of the photon. An event is the pair of detections in the two detectors. There are two ways, namely going through one hole or the other. So it's pretty clear that Feynman's framework says you should get an interference pattern.

The traditional QM framework talks about state vectors, tensor products and entanglement. Which sounds complicated, but in a simple case like this, it's actually straight forward. There's a theorem, the no-signaling theorem. It and a similar one about unitary operations says, no matter what you do to the photon, it has absolutely no effect on the electron. So the electron detector won't see interference.

This begs the question: how does the quantum eraser produce an interference pattern? It doesn't just affect the polarization of light (the which-way information), but filters half of the photons. By removing half of the photons, you can definitely affect the pattern seen on the screen. So in this case, you're using the which-way state (polarization) to affect the potentially-interfering state (whether or not the photon hits the screen).

I think the name "eraser" is an unfortunate choice, as it focuses on erasing the which-way information, rather than the filtering, and leads to exactly the confusion you had.

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It's impossible to restore the interference pattern this way.

If wave function collapse happens as a separate process from unitary evolution, then the reason is that the state collapses irreversibly when the electron's position is measured. You may be able to erase what you learned about the electron by making an uninformative measurement of the photon (triggering another collapse), but it won't restore the electron's superposition state.

If there is only unitary evolution (many-worlds), then you can't erase what you learned because unitary evolution is reversible. If you could take two different states (found-at-left-slit photon and found-at-right-slit photon) to the same state (don't-know photon), then running the physics backwards wouldn't restore the original state, which means it's impossible.

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The presence or absence of interference isn't about whether we're aware of what's happening or not. It's driven by the physical dynamics occurring in that precise moment. If "Erasing the Information" entails physically modifying any aspect of the experiment, then yes, the outcome may vary accordingly. You won't get consistent results if you obstruct or manipulate the trajectories of electrons passing through a slit experiment.

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