Generalized probability rules from a timeless formulation of Wigner's friend scenarios

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 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 Wigner and his Friend assigning different conditional probabilities to the outcome of the same measurement, given their respective observed results of a previous measurement. Different probability assignments may lead to a contradiction in the formalism, if the different predictions can be in principle compared in an experiment. In this paper, we apply the Page-Wootters mechanism to give an a priori timeless description of Wigner's friend-like scenarios, which allows Wigner and his Friend to unambiguously assign two-time conditional probabilities for the gedankenexperiment. We propose three 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 conditions on when a joint probability distribution for the measurement outcomes of Wigner and his Friend is well-defined, which single out those cases where their predictions can be compared and such probabilities have 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.