A no-go theorem for the persistent reality of Wigner's friend's perception

The paper revisits Wigner's friend, where an observer (the friend) measures a qubit inside an isolated lab while an external superobserver (Wigner) treats the entire lab unitarily. Standard analyses highlight tensions when different observers' accounts are combined. Here the authors focus on a single observer—the friend—making predictions about her own perceived outcomes at two times (before and after Wigner's intervention). They ask whether the friend's perceived outcomes can share a persistent, single classical reality across time while maintaining natural quantum assumptions (unitary single-time probabilities and linear dependence on initial states). The importance lies in showing that even without combining different observers' facts, quantum theory resists assigning a consistent joint probability over the friend's perceptions at two times, challenging how memory and identity persist in quantum scenarios.

The work is situated among recent analyses of Wigner's friend and related no-go results. Prior studies (e.g., Frauchiger–Renner) revealed inconsistencies when agents reason about others' knowledge. Other results combined observer-independent facts with locality to derive contradictions. Operational and interpretational perspectives (Everett, QBism, Bohmian mechanics) have debated the status of observed events and state updates. Experimental tests have probed local observer independence. This paper differs by targeting a single observer's two-time perceptions and by deriving a no-go result without invoking multiple agents' facts or locality, building on tools from POVMs and joint measurability.

- Formal setup: A qubit S is prepared in |ψ_S⟩ = α|↑⟩ + β|↓⟩, the friend F starts in a ready state |0⟩_F, and Wigner W in |0⟩_W. At t0 the global state is |Ψ(t0)⟩ = (α|↑⟩ + β|↓⟩)|0⟩_F|0⟩_W. At t1 the friend measures S in the z basis, yielding |Ψ(t1)⟩ = α|↑⟩|U⟩_F|0⟩_W + β|↓⟩|D⟩_F|0⟩_W. At tW Wigner measures SF in an entangled basis with outcomes |1⟩_{SF} = α|↑,U⟩ + β|↓,D⟩ and |2⟩_{SF} = β|↑,U⟩ − α*|↓,D⟩, producing a final state |Ψ(t2)⟩ that correlates SF with W's outcome states |1⟩_W, |2⟩_W. - Probabilities: Any observer assigns single-time probabilities via the Born rule p(x) = tr(Π_x |Ψ(t)⟩⟨Ψ(t)|). Mixed initial states ρ_S are treated by convex decompositions, yielding a mixed global state Σ(t) and p(x) = tr(Π_x Σ(t)). - Target question: For the friend, define events f1 (her perceived outcome at t1) and f2 (her perceived outcome at t2). The goal is to assess whether a joint distribution p(f1,f2) exists that is linear in ρ_S and has marginals matching unitary single-time Born probabilities. - Assumptions: P1: Existence of a joint distribution p(f1,f2) with standard marginalization to p(f1), p(f2). P2: One-time probabilities are assigned using unitary quantum theory (no collapse): p(f_i) = tr(Π_{f_i} |ψ(t_i)) (ψ(t_i)|). P3: p(f1,f2) depends convex-linearly on the initial qubit state ρ_S. - Proof strategy (Theorem III.1): • Define isometries V1 and V2 that map the initial qubit state to the global states at t1 and t2, respectively. • Using P2 and P3, recast single-time probabilities as p(f1) = tr(E1 ρ_S) and p(f2) = tr(E2 ρ_S), identifying POVM elements E1, E2 (on the qubit) via Heisenberg-picture pullbacks E_i := V_i^† Π_{f_i} V_i. • P1+P3 imply existence of a joint POVM {G_{f1 f2}} with the correct marginals, i.e., E1 and E2 must be jointly measurable. For two-outcome POVMs with E1 sharp, joint measurability is equivalent to [E1, E2] = 0. • Compute the commutator [E1, E2] and show it is generally nonzero for generic choices of Wigner’s measurement parameters (a, b). Hence E1 and E2 are not jointly measurable, contradicting P1–P3 simultaneously. • Note: The contradiction persists even if Wigner implements a unitary (e.g., a Hadamard on the friend–system subspace) instead of a measurement, mutatis mutandis.

- Main no-go result (Theorem III.1): In the Wigner's friend scenario, there does not exist a joint probability distribution p(f1,f2) for the friend's perceived outcomes at two times that (i) has marginals given by unitary single-time Born probabilities (P2) and (ii) depends convex-linearly on the initial state of the system (P3). Thus P1–P3 cannot all hold simultaneously for general Wigner measurement choices. - Consequences: One must relinquish at least one of: (1) linear two-time probability dependence (nonlinear modification of the Born rule), (2) using present information to predict the friend’s future observation (limiting predictability), or (3) universal validity of unitary single-time predictions for all observers. - Special cases where the commutator vanishes: • Computational basis measurement (|a|=1, b=0): memory is perfectly preserved, P(f1,f2) enforces f1=f2. • Bell-basis measurement (|a|^2=|b|^2=1/2): the friend’s memory flips with probability 1/2, independent of the initial state; this holds even when Wigner’s measurement is non-disturbing (the SF state at t1 is an eigenstate), revealing a counterintuitive state-independent 1/2 flip rate. - Conceptual point: Enforcing unitary single-time probabilities (P2) bypasses usual information–disturbance trade-offs and renders E1 and E2 non-jointly measurable in general, driving the contradiction.

The findings show that even a single observer attempting to ascribe a persistent reality to her perceptions across two times encounters conflict with core quantum features: unitary single-time predictions and linearity in the initial state. Unlike prior multi-observer no-go theorems requiring locality or cross-observer fact aggregation, this result arises from the friend's own two-time percepts. Interpretational implications are organized by which assumption is relaxed: - P1 (joint two-time facts) may be denied in Everett/many-worlds and operational views, which restrict probability assignments to decohered records and single-time memory contents; in Wigner’s friend there may be no robust record of f1 at t2. - P2 (unitary single-time predictions) can be denied by objective-collapse theories (empirically distinct from unitary QM) or via subjective state assignment (QBism), where agents update via the Born rule using personal states and may apply collapse for their own predictions. - P3 (linearity in initial state) is denied in hidden-variable theories like de Broglie–Bohm, which reproduce single-time Born statistics (P2) and maintain definite memories (P1) but entail nonlinear dependence for certain two-time scenarios. The special cases clarify when memory is preserved or randomized; notably, even non-disturbing Wigner measurements can imply a 1/2 flip if linearity is retained, challenging naive expectations about memory persistence. The work raises deeper questions about personal identity across interfering branches and the legitimacy of constructing joint probabilities across times in quantum settings.

The paper establishes a no-go theorem demonstrating that a consistent, linear, joint two-time probability for the friend's perceived outcomes cannot coexist with unitary single-time marginals in Wigner’s friend scenarios. This advances foundational understanding by shifting the tension from multiple observers' facts to a single observer’s temporal perceptions. The result pressures interpretations to relinquish one of P1–P3 and clarifies how Everett, QBism, collapse, and Bohmian frameworks navigate the conflict. For practical, decohered conditions, predictive use of present information remains effective, but fundamentally the theorem limits persistent classical reality of memory under unitary quantum dynamics. Future work includes deriving explicit two-time probability rules in Bohmian mechanics for this setup, exploring operational conditions under which reliable records permit joint assignments, and further analyzing identity and agency across times in quantum interference scenarios.

- The analysis assumes idealized, fully unitary dynamics and control over Wigner’s measurement/operations in an isolated lab; real experiments have decoherence that may restore effective joint descriptions. - The friend’s outcomes are modeled with two relevant states; additional outcomes are ignored when they never occur in the protocol. - The contradiction depends on generic Wigner measurement choices; special bases avoid it but can yield counterintuitive conclusions (e.g., 1/2 memory flip) still contingent on assumptions. - The result is primarily conceptual; no direct empirical test is provided for the two-time joint probabilities, especially as reliable records of f1 may not persist to t2 in this scenario. - The theorem relies on linearity in the initial state for two-time probabilities (P3); dropping it requires specifying alternative, possibly nonlinear, two-time rules not derived here.