Can quantum theory consistently describe the use of itself?

Heads, tails, or something else?

In my previous post I described the Wigner-Deutsch thought experiment. The thought experiment showed that the friend was able to send a message to Wigner indicating that she had performed a measurement and observed a definite result.

Now once the experiment has been completed, including the reverse unitary transformation there described, what result will the friend predict that Wigner will see?

On a non-unitary interpretation (which involves an objective collapse and a single definite result), she will predict a 50% chance of seeing heads (since the definite result obtained in the experiment is now in superposition again). Whereas on a unitary interpretation, she will predict a 100% chance of seeing heads (since her lab which was in superposition during the experiment has now been restored to its original state).

A similar reasoning step occurs in the Frauchiger-Renner thought experiment where Alice (one of the friends), on account of seeing tails, concludes that Bertrand (one of the Wigners) will see \(\boldsymbol{|+\rangle}\).

But that is only true if her measurement is non-unitary. However if her lab is in superposition with respect to Bertrand (and Alice) then her lab will subsequently be reversed to its original state. As such, she can't conclude what Bertrand will see from her own measurement. Similarly, Ada can't conclude what Bob and Alice saw since, at the time of Ada's measurement, the friends' measurements will have been reversed.[1] Thus if Ada measures \(\boldsymbol{|-\rangle}\) she can only predict, per quantum theory, that Bertrand could also measure \(\boldsymbol{|-\rangle}\) with probability \(\frac{1}{12}\).

So, on the assumption of unitarity, there is no inconsistency.

How about on the assumption of non-unitarity? In this case, Ada can conclude that Bertrand will see \(\boldsymbol{|+\rangle}\). Again there is no inconsistency since quantum theory doesn't apply to this case - the state would have collapsed at an earlier time.

Inconsistency only arises when combining Ada's inferential reasoning (which assumes non-unitarity) and the prediction from quantum theory (which assumes unitarity). So the solution to the paradox is to recognize that measurements will be either unitary or non-unitary, but not both at the same time (i.e., depending on whether one is inside or outside the lab).

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[1] As an analogy, consider a qubit that has been been measured as \(|0\rangle\) in the \(\{|0\rangle,|1\rangle\}\) basis. Suppose the qubit is then subsequently measured as \(|+\rangle\) in the \(\{|+\rangle,|-\rangle\}\) basis. If the qubit is again measured in the \(\{|0\rangle,|1\rangle\}\) basis, there is a 50% chance that the result will be \(|1\rangle\). That is, the original measurement result has been lost and can no longer be used in our reasoning.