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msumm21

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- 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##.

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##.