Aumann's theorem beyond ontology: quantum, postquantum, and indefinite causal order

1. Problem Statement

Aumann's Agreement Theorem is a foundational result in Bayesian epistemology stating that two rational agents with a common prior cannot "agree to disagree." If their posterior beliefs regarding an event become common knowledge (i.e., each knows the other's belief, knows that the other knows, ad infinitum), their posterior probabilities must be identical.

The Core Issue:
Standard formulations of Aumann's theorem rely on an ontological framework:

• They assume a pre-existing "state of the world" ($\Omega$).
• Agents possess partitions of this state space based on their observations.
• Probabilities are derived by conditioning on these partitions within a fixed underlying reality.

The Challenge:
In quantum mechanics and beyond (postquantum theories), the assumption of an objective, observer-independent state of the world is controversial or potentially invalid (e.g., due to non-commuting measurements or the Wigner's friend paradox). Previous literature suggested that Aumann's theorem might fail in quantum settings, particularly when measurements do not commute, or that it requires counterfactual reasoning that leads to "singular disagreement." The authors ask: Can Aumann's theorem be formulated without assuming an underlying ontic state space, relying solely on operational observations?

2. Methodology

The authors propose an Operational Agreement Theorem that shifts the focus from the "state of the world" to the space of measurement outcomes.

• Reframing the Probability Space: Instead of defining a prior over a state space $\Omega$, the agents share a prior probability distribution $p(i, j, k)$ over the joint outcomes of three measurements:
  • Alice's outcome: $i \in I$
  • Bob's outcome: $j \in J$
  • The event of interest: $k \in K$
• Bayesian Conditioning: Agents update their beliefs using standard Bayesian conditioning on their observed outcomes.
  • Alice's posterior: $p_A(k|i) = p(i, k) / p(i)$
  • Bob's posterior: $p_B(k|j) = p(j, k) / p(j)$
• Defining Common Knowledge Operationally: Common knowledge is defined recursively over the sets of outcomes. If Alice observes $i$, she knows Bob's outcome $j$ must be in a specific set $B_n$ if she knows his posterior. This creates a sequence of sets $A_n$ and $B_n$ representing the intersection of knowledge.
• Generalization: The framework does not specify how the joint distribution $p(i, j, k)$ is generated. It only requires that a valid joint probability distribution exists for the outcomes. This allows the theorem to apply to:
  • Classical theories (marginalizing over hidden states).
  • Quantum mechanics (using instruments and trace rules).
  • Indefinite Causal Order (using process matrices).
  • Postquantum theories (Generalized Probabilistic Theories).

3. Key Contributions

1. Ontology-Free Formulation: The paper demonstrates that the existence of a "true state of the world" is not essential for Aumann's theorem. The theorem holds purely based on the consistency of probabilistic predictions and Bayesian conditioning on a joint outcome space.
2. Extension to Non-Commuting Measurements: Unlike previous attempts that restricted the theorem to commuting quantum measurements, this operational version applies to non-commuting measurements. The authors prove that as long as a joint probability distribution over the outcomes exists (which quantum mechanics provides via the Born rule), agents cannot agree to disagree.
3. Application to Indefinite Causal Order: The theorem is extended to scenarios where the causal order of events is indefinite (superposition of orders). Using the Process Matrix formalism ($W$-matrix), the authors show that even when the causal order of Alice's, Bob's, and a third party's measurements is not fixed, the agreement theorem holds provided a joint probability distribution can be computed.
4. Resolution of Literature Contradictions: The authors clarify apparent contradictions in recent literature (e.g., Contreras-Tejada et al., Díaz et al.):
   • They argue that previous "counterexamples" often relied on counterfactual reasoning (assigning probabilities to measurements that were not performed) or non-standard update rules (treating post-measurement states as superpositions rather than mixtures).
   • Under standard operational rules where agents condition only on actual observed outcomes, the agreement theorem remains robust.

4. Key Results

• Theorem 2 (Operational Agreement Theorem): If rational agents share a prior distribution over measurement outcomes and their posteriors are common knowledge, their posteriors must be equal ($q_A = q_B$).
• Proof Logic: The proof relies on the fact that if common knowledge exists, the sets of outcomes consistent with the agents' knowledge ($A^*$ and $B^*$) must have probability 1 conditional on each other. This forces the conditional probabilities of the event of interest to be identical.
• Quantum Example: The authors construct a specific example involving projective measurements on a 4-dimensional Hilbert space with non-commuting bases. They show that for specific initial states and outcomes, common knowledge arises, forcing the agents to assign identical probabilities to the event $E$.
• Postquantum Validity: The theorem holds for any theory (including Generalized Probabilistic Theories or Constructor Theory) that allows for the calculation of joint probabilities for multiple measurement outcomes.

5. Significance and Limitations

Significance:

• Foundational Robustness: It establishes that the impossibility of rational disagreement is a feature of probabilistic consistency rather than a feature of a specific physical ontology (like classical realism).
• Unification: It unifies classical, quantum, and postquantum agreement under a single operational framework, removing the need for "hidden variables" or objective states to explain why agents agree.
• Clarification of Quantum Paradoxes: It provides a clear boundary for where quantum mechanics behaves classically regarding information sharing: as long as joint probabilities are well-defined, "agreeing to disagree" is impossible.

Limitations and Future Directions:

• Wigner's Friend Scenarios: The authors identify the Wigner's friend thought experiment as the primary potential failure point. In these scenarios, different observers may not be able to assign a joint probability distribution to their respective outcomes (due to the lack of a shared, observer-independent reality). If a joint distribution $p(i, j, k)$ cannot be meaningfully defined, the operational agreement theorem ceases to apply. This suggests that Wigner's friend scenarios represent a genuine breakdown of the standard agreement framework, rather than a failure of the theorem itself.

Conclusion

Cepollaro and Di Biagio successfully decouple Aumann's Agreement Theorem from classical ontology. By reformulating it in terms of joint measurement outcomes, they prove that rational agents cannot agree to disagree in any physical theory (classical, quantum, or postquantum) that supports a well-defined joint probability distribution. This reinforces the view that the consistency of probabilistic predictions is a fundamental constraint on rationality, independent of the underlying nature of reality, with the notable exception of scenarios involving observer-dependent facts (Wigner's friend).