Double slit with erasure 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 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 in 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 electrons 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 anwer?
 A: 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.
A: 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.
