I'm pretty certain that the answer to the question in the title is a no, but I don't understand why. I have some basic misunderstanding of quantum processes that I’d like clarified in the form of asking questions about a hypothetical scenario.
So, let’s say we have a pair of entangled photons…
1) Is it possible to prepare an entangled pair such that it can be known that, for example, the photon going down a certain path will be measured 25% of the time to be horizontally polarized and 75% likely to be vertically polarized? I guess this would be some sort of quantum decoherence? Partial collapse?
2) If that is possible, then would it also be possible to measure that photon’s diagonal polarization in order to "reset" the horizontal-vertical probability back to 50-50? I've seen a trick done where you take some polarizing filters and stack them horizontal on top of vertical, so that no light gets through. But when you slide a diagonally oriented filter between them, some light does get through, because the horizontal/vertical polarization information gets "reset" when measured against the diagonal filter.
3) If these two things are possible, then wouldn’t it be possible to send information along the entangled pair? If you sent bursts of 1000 photons and were able to measure them all, and found that the first burst had ~750 vertically polarized photons but the second burst had ~500 vertically polarized photons, then wouldn’t you know that the second burst had their diagonal measurements taken? Or would that not work because the extra measurements "break" the entanglement..?
4) Along with that, how many times can the photon pairs be measured before there are no longer entangled? Just once? Or maybe just once per state/qubit/aspect (such as polarization angle)? I understand that measuring the photon would necessarily change it (like with the polarization filter trick), but then how do scientists measure the polarization angles of two photons like in experiments of Bell’s inequality?
I know that somewhere along the line, what I've described cannot be possible. If it were, then you could easily send information faster than light. E.g., say you had a photon-entanglement-generator-and-probability-fixer station in between two planets, 1 light year from the planet Foo and .99 light years from the planet Bar. After a year's time, after the photon stream has propagated to both planets, scientists on Bar could measure diagonally polarization of 1000 photons to encode a 1 or not measure to encode a 0. Then scientists on the planet Foo could measure vertically polarity and decode the 1s and 0s based on the statistical readings. The Bar scientists could encode that such a message and the Foo scientists would "instantly" receive the message. Except it would actually be from the past, since Foo is 1.99 years outside the Bar light cone. So since this is impossible, what am I misunderstanding?
Thanks for answering, and thanks for putting up with yet another clueless but curious layman.