Relational QM Example, Contradiction?

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msumm21
TL;DR Summary
I started reading about the “relational” interpretation of quantum mechanics, but it at first seems contradictory. An example of such a contradiction is provided. I'd like to understand where I'm going wrong.
I started reading about the “relational” interpretation of QM (RQM). I’m stuck on what appears to be a contradiction. The simplest way I can think to explain it is with a slight complication added to the Wigner’s friend experiment, adding an extra operation after the friend F measures the qubit. Maybe someone can see where I'm going wrong?

Wigner’s friend F measures a qubit, initially ##|0\rangle + |1\rangle##, revealing either state ##|0\rangle## or ##|1\rangle##. If Wigner doesn’t know this result, my understanding is that RQM says the state of the qubit and F is the superposition ##|F_0 0\rangle + |F_1 1\rangle## (according to Wigner). At this point (normal Wigner’s friend), I understand F and Wigner are using different states, but don’t yet see a potential contradiction. However, the next step is where things don’t make sense to me.

Let’s put this composite system (F and the qubit) through the operation ##A## (defined below). The result afterward will always be ##A(|F_0 0\rangle + |F_1 1\rangle) = |F_0 0\rangle## (according to Wigner). However, the friend, using state ##|F_0 0\rangle## (or ##|F_1 1\rangle##), thinks the state ends up ##A|F_0 0\rangle = |F_0 0\rangle+|F_1 1\rangle## (or ##A|F_1 1\rangle = |F_0 0\rangle-|F_1 1\rangle##). So, from what I gather, RQM says Wigner would always measure the final state of this system to be ##|F_0 0\rangle##, but F concludes he measure 1 half the time. So W and F get different results (half the time) from the same experiment, right? Is this OK somehow?

Operation A maps basis vectors like this:
##A|F_0 0\rangle = |F_0 0\rangle+|F_1 1\rangle##,
##A|F_1 1\rangle = |F_0 0\rangle-|F_1 1\rangle##,
##A|F_0 1\rangle = |F_0 1\rangle##, and
##A|F_1 0\rangle = |F_1 0\rangle##.

msumm21
Thanks for the link. I don't think it resolves my problem though. In the context of my example, they assume the qubit and F interact with an environment E and decohere before any subsequent operation A. I agree this would resolve the apparent contradiction, but presumably RQM needs to work even without this extra step of decoherence, right?

*now*
You're welcome. Given all the conditions and so on, like that decoherence is relative to a system, that a fact relative to F can become true for W, without W interacting with F, or subtleties such as allowing for "another system W′ that couples differently to these systems might still be able to detect interference effects", I don’t see a problem with the description of the interpretation.

msumm21
You're welcome. Given all the conditions and so on, like that decoherence is relative to a system, that a fact relative to F can become true for W, without W interacting with F, or subtleties such as allowing for "another system W′ that couples differently to these systems might still be able to detect interference effects", I don’t see a problem with the description of the interpretation.
Do you agree that F and W get inconsistent results, but think that's OK, or do you not agree that F and W can get inconsistent results?

In found this paper, link below, where Rovelli seems to have pointed out the problem (page 4, right column, after “Hypothesis 1”). Speaking of such interference effects, he says “these discrepancies are likely to be minute, as shown by the beautiful discovery of the physical mechanism of decoherence.” I feel like I must be missing something, is it true that RQM only works in such scenarios after complete decoherence? If so, why would it be taken seriously as an interpretation?

https://arxiv.org/abs/quant-ph/9609002

*now*
Yes, I think in a context in which the conditions are met, Wigner and Friend agree. I think rather than requirement for complete decoherence, an approximate nature is described e.g., “ These observations show that decoherence does not imply that there is a perfectly classical world of absolute facts, although it does explain why (and when) we can reason in terms of stable, hence approximatively classical, facts.2”, and there is a limit case absent decoherence, so, I don’t think the follow on question applies.

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