The Wigner's friend thought experiment, and its variations, has sparked renewed interest due to challenges it poses to fundamental assumptions of quantum theory. This experiment involves an observer (the friend) performing a quantum measurement on a system within a sealed laboratory, while a superobserver (Wigner) observes both the system and the friend. The counter-intuitive situation arises where Wigner describes the friend in a quantum superposition of observing different outcomes, while the friend perceives a definite outcome. Recent works have shed new light on this, often focusing on the problematic nature of treating observational knowledge of other agents as one's own. The Frauchiger-Renner argument, for instance, highlights ambiguities in nested reasoning about information held by different agents within the quantum theoretical framework. Other studies have shown no-go results by combining observations from multiple observers within a shared classical reality. This paper deviates from this approach, focusing on a single observer making predictions about their own observations at two different times. The core argument is that if the observer is subjected to a measurement by a superobserver between these two times and uses unitary quantum mechanics, a conflict arises with the linear dependence of quantum mechanical probabilities on the density operator. This challenges the idea that information from the past persists unchanged into the present, even considering subjective uncertainty, within the framework of quantum theory.

The paper reviews existing literature on Wigner's friend thought experiments, highlighting various approaches to resolving the paradox. It mentions the Frauchiger-Renner argument, which points to ambiguities in nested reasoning about observational knowledge held by multiple agents in a quantum setting. This ambiguity is analogous to the QBist interpretation. The paper also notes the emergence of no-go theorems that arise when combining the observations of multiple observers, assuming those observations belong to a single classical reality. These works often incorporate assumptions such as the treatment of other observers' outcomes as facts of the world or the absoluteness of observed events. This paper differentiates itself from these approaches, focusing on a single observer and their predictions across two different times. The implications of the Frauchiger-Renner argument are reinterpreted, suggesting a focus on the invalidity of inferences based on a quantum state assigned at a certain time, particularly when influenced by subsequent measurements.

The paper develops a formal no-go theorem to address the persistent reality of Wigner's friend's perception. The methodology utilizes a unitary formalism within quantum mechanics, avoiding the collapse postulate. The scenario involves a friend (F) measuring a qubit (S), with outcomes U or D, within a sealed laboratory, while Wigner (W) performs a subsequent measurement on both the system and the friend. The initial state of the system is described as |Ψ(t₀)| = (α|↑⟩ₛ + β|↓⟩ₛ)|0⟩F|0⟩W. The friend's measurement at t₁ leads to |Ψ(t₁)| = (α|↑⟩ₛ|U⟩F + β|↓⟩ₛ|D⟩F)|0⟩W. Wigner's measurement at tW, in an entangled basis, results in a final state |Ψ(t₂)| at t₂ > tW. Probabilities are calculated using the Born rule, p(x) = tr(Πₓ|Ψ(t)⟩⟨Ψ(t)|), where Πₓ is a projector onto the state where the observer sees outcome x. The analysis is extended to mixed states ρₛ using a decomposition ρ = λ|ψ⟩⟨ψ| + (1 − λ)|φ⟩⟨φ|. The core of the theorem rests on three assumptions: P1 – the existence of a joint probability distribution p(f₁, f₂) for the friend's perceived outcomes at times t₁ and t₂ with consistent marginals; P2 – one-time probabilities are assigned using unitary quantum theory without state update rules; and P3 – the joint probability p(f₁, f₂) depends linearly on the initial state ρₛ. The proof involves the construction of POVMs for the friend's observations at t₁ and t₂, showing that these POVMs are not jointly measurable for a general choice of Wigner's measurement basis. The non-joint measurability demonstrates the incompatibility of the three assumptions in Wigner's friend scenario.

The key finding is the no-go theorem that proves the incompatibility of three seemingly reasonable assumptions concerning probability assignments in the Wigner's friend experiment. The theorem demonstrates that it's impossible to simultaneously maintain: 1) a joint probability distribution for the friend's perceptions at two different times with consistent marginals; 2) the use of unitary quantum theory for single-time probability assignments without the state-update rule; and 3) a linear dependence of joint probabilities on the initial quantum state. The proof is based on showing the non-joint measurability of POVMs representing the friend's observations at two different times. Even when Wigner performs a non-disturbing measurement, leading to no change in the quantum state, the friend's perceived outcome still shows a 50% probability of flipping, a counter-intuitive result that highlights the conflict between these assumptions. The paper analyzes specific cases where the assumptions can be satisfied (e.g., Wigner measuring in the computational or Bell basis), revealing interesting implications regarding the persistence of the friend's memory. In the Bell basis measurement, the friend's memory of their outcome has a 50% chance of being flipped, even if Wigner's measurement doesn't change the quantum state. This counter-intuitive result suggests a conflict with the assumption of linear dependence on the initial state.

The no-go theorem challenges several fundamental aspects of quantum mechanics and its interpretations. It highlights the difficulties in consistently combining the unitary evolution of quantum systems with the description of observers' experiences and their memories. The impossibility of satisfying the three assumptions simultaneously suggests that we must reconsider core principles within quantum theory. The results force a choice between modifications to the Born rule, limitations on the predictive power of quantum mechanics, or a denial of the universality of unitary quantum mechanics for single-time predictions. The paper explores how different interpretations of quantum mechanics might deal with this incompatibility, suggesting strategies for modifying or rejecting each assumption. The Everett interpretation, for instance, might reject the possibility of assigning a joint probability to the friend's observations across time. QBism might reject the assumption of unitary time evolution for probability assignments. Bohmian mechanics might violate the linearity assumption. The implications for the concept of an observer's identity are also discussed, suggesting the possibility that the friend at different times might be considered distinct agents.

The paper concludes that treating the friend's memory of a measurement outcome as a persistent reality across time conflicts with core aspects of quantum mechanics, as revealed by the no-go theorem. This incompatibility forces a reevaluation of fundamental assumptions in quantum theory and its interpretations, suggesting potential revisions to probability assignments, predictive power, or the scope of unitary quantum mechanics. Future research should delve deeper into how specific interpretations of quantum mechanics accommodate this no-go theorem and explore the implications for the persistence of identity and the observer's role in quantum mechanics.

The study's primary limitation is the reliance on three assumptions that might be deemed debatable in certain interpretational frameworks. The validity of these assumptions depends heavily on one's chosen interpretation of quantum mechanics. While the authors provide motivations for these assumptions, their rejection opens avenues for alternative interpretations and resolutions of the Wigner's friend paradox. The analysis is also limited to a specific type of Wigner's friend experiment, and generalizations to more complex scenarios might reveal further nuances.