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.

中文翻译：

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来自 Wigner 朋友场景的永恒公式的广义概率规则

量子测量问题可以看作是量子力学规定的两种替代动力学之间的张力：波函数的幺正演化和测量发生瞬间的状态更新规则（或“崩溃”）。臭名昭著的 Wigner 的朋友 gedankenexperiment 构成了一个矛盾的场景，其中不同的观察者（一个被另一个观察到）以不同的方式描述一个相同的交互，一个 - 朋友 - 通过状态更新，另一个 - Wigner - 统一。这可能导致 Wigner 和他的朋友为相同的后续测量结果分配不同的概率。在本文中，我们应用 Page-Wootters 机制 (PWM) 作为对 Wigner 类似朋友的场景的永恒描述。我们表明需要修改在 PWM 内分配两次条件概率的标准规则，以处理 Wigner 的朋友 gedankenexperiment。我们确定了此类修改规则的三个主要定义，以分配两次条件概率，所有这些定义都归结为非 Wigner 朋友场景的标准量子理论。然而，当应用于 Wigner 的朋友设置时，每个规则分配不同的条件概率，可能以不同的方式解决概率分配悖论。此外，一项规则对 Wigner 和他的朋友的测量结果的联合概率分布何时定义明确施加了严格的限制，这会挑出那些 Wigner 的测量不会干扰朋友的情况。s 记忆和这种概率在可收集统计方面具有操作意义。有趣的是，相同的限制保证了所述测量结果满足一致性历史框架的一致性条件。