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Hmm interesting. I may have been mistaken about the electrons only being entangled in a single direction. I thought that if you prepared a pair of electrons in state
1/sqrt(2) (|z+z-> + |z-z+>)and then measured it in y there would be no correlation, but based on: https://physics.stackexchange.com/questions/700218/intuition-for-results-of-a-measurement-of-entangled-spinshttps://physics.stackexchange.com/questions/385477/what-is-the-quantum-state-of-spin-1-2-particle-in-y-direction
if I had done the 90° rotation properly, the math works out such that the electrons would still be entangled in the new y+ basis! There is no way to only entangle them in z alone - if they are entangled in z they are also entangled in x and y. My math skills were 20 years rusty, sorry!
I still think my original proposition, that in the DCQEE under Copenhagen, an observation that collapses one photon, collapses the other photon to a sub-superposition, can be salvaged. In the second stackexchange link we are reminded that for a single electron, the superposition state
1/sqrt(2) (|y+> - |y->)is the same as|z+>state! They describe the same wavefunction psi, expressed in different basis: (y+,y-) vs. (z+,z-). When we take a single electron in superposition1/sqrt(2) (|z+> + |z->)and measure it in z, and it collapses to, say, z+, we know that it is a pure state in z basis, but expressed in y basis it is now a superposition of1/sqrt(2) (|y+> - |y->)! Indeed if we measure it now in y, we will get 50% y+ and 50% y-.So in DCQEE when you collapse the first photon into a single position on the screen, the twin photon does collapse, but its basis is not expressed in terms of single positions! It's some weird agglomeration of them. If you were to take that "pure" state and express it in terms of position basis, you would get a superposition of, say, 80% path A and 20% path B.
Well, if the second photon is in a new, weird superposition then the first photon must also be in the same new, weird superposition. Again, I don't that's compatible with Copenhagen given that the first photon no longer exists.
Note by the way that 50% y+ and 50% y- is how all photons start. So if that's also the final state then there is no reason for it to prefer any detector over the others.
Yeah, but when you start with a 50% z+ / 50% z- electron, and you measure it and get say z+, it is now 100% z+, right? If you measure it again, you will always get z+. And then you give a bunch of them to your buddy with an identical lab and an identical Stern-Gerlach apparatus and and they say "hey, I measured your electrons that you said were 100% z+, and I'm getting 50% z+ 50% z-". And you say "dude! your lab is in China! your z+ is my y+! you have to do coordinate rotation and basis substitution! if you look at my pure electron in your sideways basis, it's in superposition for you".
When the first photon hits the screen, the basis is the screen basis. Each position on the screen - 1.4, 1.5, 1.6, etc - is an eigenvector and the first photon collapses to one of those eigenvectors. The second photon collapses too, but you are wrongly equating the positions on the screen and positions on paths A/B as if they are in the same basis. They are not! You were just misled to think they are the same basis because they are both named "position", but they are as different as the z+ axis in America is different from z+ axis in China.
The second photon collapses into the screen basis eigenvector 1.5 but that 1.5 does not correspond to any single location on path A or path B. If you do the basis substitution from screen basis into path basis, you get something like 80% path A and 20% path B (and something weird with the phases too I bet). Does that sound accurate?