Abstract
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.
Highlights
Standard quantum theory features two dynamical processes: the so-called “collapse of the wave function” that occurs during a measurement and the unitary evolution describing the propagation of the wave function in the absence of measurements
Since Eq (7) is identical to the probability assigned to the measurement outcome m given the state of the system is |ψS(t) by the Born rule in the standard formulation of quantum theory, we are justified in identifying the conditional state in Eq (5) as the standard time-dependent wave function
We have presented three generalizations of the standard rule to assign probabilities to consecutive quantum measurements in a Wigner’s friend scenario using the Page-Wootters mechanism
Summary
Standard quantum theory features two dynamical processes: the so-called “collapse of the wave function” that occurs during a measurement and the unitary evolution describing the propagation of the wave function in the absence of measurements. [5, 6, 7], the seemingly natural assumptions that quantum theory is universal, that different observers can each apply either of the two dynamical processes of the theory with respect to their description, and that the reasoning of different agents about one another can be freely combined to make statements about an objective reality lead to contradictions for Wigner’s friend setups. These contradictions can be regarded as resulting from an observer dependent application of the state-update rule and involve inferences about the outcomes different observers obtain at different times. In case of the last rule, these conditions single out those settings for which Wigner’s measurement does not disturb the Freind’s memory and, the respective probabilities correspond to collectible statistics for the Friend
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