The quantum measurement problem can be regarded as the tension between the two alternative dynamics prescribed by quantum mechanics: the unitary evolution of the wave function and the state-update rule (or "collapse") at the instant a measurement takes place. The notorious Wigner's friend gedankenexperiment constitutes the paradoxical scenario in which different observers (one of whom is observed by the other) describe one and the same interaction differently, one –the Friend– via state-update and the other –Wigner– unitarily. This can lead to Wigner and his friend assigning different probabilities to the outcome of the same subsequent measurement. In this paper, we apply the Page-Wootters mechanism (PWM) as a timeless description of Wigner's friend-like scenarios. We show that the standard rules to assign two-time conditional probabilities within the PWM need to be modified to deal with the Wigner's friend gedankenexperiment. We identify three main definitions of such modified rules to assign two-time conditional probabilities, all of which reduce to standard quantum theory for non-Wigner's friend scenarios. However, when applied to the Wigner's friend setup each rule assigns different conditional probabilities, potentially resolving the probability-assignment paradox in a different manner. Moreover, one rule imposes strict limits on when a joint probability distribution for the measurement outcomes of Wigner and his Friend is well-defined, which single out those cases where Wigner's measurement does not disturb the Friend's memory and such a probability has an operational meaning in terms of collectible statistics. Interestingly, the same limits guarantee that said measurement outcomes fulfill the consistency condition of the consistent histories framework.
In our work, we have argued that a way to clarify the conceptual issues of the measurement problem in a Wigner's friend scenario is to eliminate time (in a classical sense, that is, a parameter which always increases) from the description. We have shown that in this way, Wigner and the friend remove the which-dynamics ambiguity because they always agree on a single timeless state of the whole physical situation. However, even if they always agree, they can use different rules to predict probabilities for measurement results. We have shown that some probability rules give the probabilities provided by unitary dynamics, while others lead to probabilities equal to those given by "collapse“ dynamics. This means, depending on which rule Wigner and his friend use, they will either agree on the probabilities for unitary dynamics or those for collapse dynamics. In our framework, the ambiguity between the two dynamics is thus pushed back to the choice of the probability-assignment rule, but once Wigner and the friend agree on a rule (ideally based on some physical arguments or evidence), there cannot be disagreement on the probabilities.
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