Wigner's friend scenarios: on what to condition and how to verify the predictions

TL;DR

The paper addresses whether a single objective quantum state can describe a Wigner–friend scenario across observers. It develops an information-bubble framework and a bubble-switching game to test when an observer should adopt another bubble’s state assignment for prediction, incorporating a boundary interaction $U_ ext{int}$ parameterized by $ heta$ to modulate information leakage. The main contributions are formalizing bubbles as information-theoretic contexts, introducing a symmetric game that reveals when cross-bubble state assignments become advantageous, and showing that relative objectivity can emerge for both the friend and Wigner under suitable conditions (e.g., higher $ heta$ or specific measurements). The findings challenge the notion that Wigner always has superior predictive power, clarifying interpretational possibilities and suggesting a refined, context-dependent notion of objectivity with potential implications for quantum foundations and quantum reference-frame frameworks.

Abstract

Wigner's friend experiment and its modern extensions display the ambiguity of the quantum mechanical description regarding the assignment of quantum states. While the friend applies the state-update rule to the system upon observing an outcome of her measurement in a quantum system, Wigner describes the friend's measurement as a unitary evolution, resulting in an entangled state for the composite system of the friend and the system. In this respect, Wigner is often referred to as a "superobserver" who has the supreme technological ability to keep the friend's laboratory coherent. As such, it is often argued that he has the "correct" description of the state. Here we show that the situation is more symmetrical than is usually thought: there are different types of information that each of the observers has that the other fundamentally cannot have - they reside in different "bubbles" (in Calvalcanti's terminology). While this can explain why the objectivity of the state assignment is only relative to the bubble, we consider more elaborated situations in the form of a game in which the players can switch between bubbles. We find that, in certain circumstances, observers may be entitled to adopt and verify the state assignment from another bubble if they condition their predictions on \textit{all} information that is in principle available to them.

Figures (4)

• Figure 1: Wigner and his friend assign in general different states to the same physical systems, i.e., they live in different “bubbles". In the friend's bubble a measurement of a qubit is performed and an outcome is obtained. We assume that there exists an interface at the “boundary" of the two bubbles where the environmental states of the friend's and Wigner's bubble unitarily interact. Depending on the unitary $\hat{U}_{\text{int}}$, information about the results of the friend's measurements may or may not be encoded in the Wigner's environment, making the descriptions of the friend and the Wigner compatible (hence merging the two bubbles in one) or not. In the extreme case of the unitary (\ref{['collmeas1']}), there is no information about the friend's outcome in Wigner's environment.
• Figure 2: The schematic illustration of the game protocol in the case that measurement $M_W$ in Wigner's bubble is chosen to be performed. On the basis of quantum state assignments in their respective laboratories, the two players make their predictions and submit them to the responsible referee R$_W$ in Wigner's bubble. The probabilistic prediction of the friend contains no information about the outcome of the qubit observed in her laboratory. Consequently, its communication to the referee has no effect on the state of the laboratory as described by Wigner. The outcome of the $M_W$ measurement is in accordance with the prediction of Wigner, and in repeated trials, a discrepancy from the prediction of the friend can be observed.
• Figure 3: The schematic illustration of the game protocol in the case that measurement $M_F$ in the friend’s bubble is chosen to be performed. The two players make their predictions on the basis of their state assignments and submit them to the responsible referee R$_F$ in the friend’s bubble. The prediction of Wigner can be sent inside the friend's laboratory with no additional effects. The outcome of the measurement is consistent with the prediction of the friend, and in repeated trials, a discrepancy between the observed relative frequencies and the prediction of Wigner can be observed.
• Figure 4: Top panel: Two examples of Wald's sequential probability ratio test for $\theta = 0$ (case in the main text). The blue dots represents the log-likelihood ratio for sequence of consecutive $\omega_0$ outcomes, reaching the acceptance threshold of Wigner's prediction (dashed line) for $\epsilon= 10^{-3}$ at run $n=10$. The green dots represent instead the log-likelihood ratio in the eventual case of a sequence of 4 consecutive $\omega_0$ values followed by $\omega_1$, leading to the acceptance of the friend's prediction at run $n=5$. Bottom panel: Minimum number of runs $n_0$ leading to outcomes $\omega_0$ in the measurement $M_W$ needed to validate Wigner's prediction as a function of the encoding parameter $\theta$ in the range $[0, 1]$ for two different values of the failure probability $\epsilon$.