Okay, this was an interesting answer, and I will have to think about this for a moment. But first let me just clarify: so there is actually no way of talking about the state of a particle compared to some reference frame, only the state of the particle compared to some other particular object?

We can only talk about the state of a particle in the reference frame of a particular observer, and then the state concerns the probability distribution for the measurement results which that observer could obtain.

Charles, right, so the state of a particle only makes sense compared to some observer or object, and when I say "the wave function has collapsed", then I should really specify who it collapsed with repsect to?

Yes, that is correct. Unfortunately a great number of physicists, and especially popularisers, do not understand this and consequently they spread a great deal of confusion.

Yeah, I see. The only place where I have ever seen this mentioned before is in one of Carlo Rovelli's popular science books. But I haven't even seen it being mentioned in my quantum physics text book.

Yes, quantum physics texts books duck out of this kind of issue, and focus on experimental results. A lot of physicists just want to avoid controversy. One really has to make a devoted study at postgraduate level into mathematical foundations of quantum mechanics if one is interested in interpretation.

This answer is a bit odd. If in the state $|0>|0>+|1>|1>$ Alice measures 0 on her qubit, so will Bob do when he measures his. Bob's wavefunction for qubit B will collapse at the moment Alice measures her qubit.

@proton, it would indeed be odd if Bob's wavefunction collapsed at the moment of Alice's measurement, since that would mean that for different observers it also collapses both before and after Alice's measurement. However it is not odd one understands that wave functions are just probabilities. Probabilities collapse due to change information. Bob has no change in information from Alice's measurement, so his wave function does not collapse.

Charles, one thing I do find odd, though, is that your answer seems to be referring to relational quantum mechanics, not the conventional Copenhagen interpretation. But surely the Copenhagen interpretation must have something to say here too, as I can't imagine that I am the first who thought about this thought experiment, so somebody must have surely come up with a solution, right?

@FelisSuper. There are many forms of the Copenhagen interpretation. In Interpreting the Quantum World Jeffrey Bub separates off the orthodox, or Dirac-von Neumann interpretation, which has largely supplanted the traditional Copenhagen interpretation of Bohr, but is still often called Copenhagen. The main difference is that in the orthodox interpretation, complementarity, or wave-particle duality, has no role. Collapse applies to probabilities, not to physical waves. My answer is entirely within the orthodox form of the Copenhagen interpretation.

@CharlesFrancis Because of the entanglement it is impossible for Bob to measure 1 when Alice measures 0. So Bob's wavefunction must collapse when Alice measures here qubit.

@proton, It is impossible for Bob to know anything about Alice's measurement, so his probability does not change, and nor does his wavefunction.

@CharlesFrancis You imply that bob can measure 1 When Alice measures 0 which is impossible. I don't say anything about what Bob (or Alice) knows, I only say what he can measure.

proton, as far as I have understood it so far, Bob's wave function can't collapse at the moment of Alice's measurement, because Bob doesn't know about the measurement. Therefore, as far as he is concerned, there is still a 50-50 chance that he will measure spin up/spin down. But of course, ALICE knows that Bob will measure spin down, but Bob doesn't know that.

So as far as Bob is concerned, neither Alice's nor his wave function collapses at the moment of Alice's measurement.

@proton, please read the answer to the question. Entanglement, and quantum mechanics generally, concerns what people can know. It does not involve supernatural properties like instantaneous action at a difference as you seem to think. You may also find the second paragraph of en.wikipedia.org/wiki/Copenhagen_interpretation helpful

@CharlesFrancis I read your answer before and I understands all of this issues. What I don't understand is that you answer appears to allow a situation where Bob measures 1 and Alice measures 0 although this should not happen.

@proton, it remains the case that neither can say what the other measured until later. The question is not even meaningful until the results are brought together. As for what it means, I have given a discussion in my 2nd book, but it is outside the scope of the present question.