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

Carlo Cepollaro, Andrea Di Biagio

Abstract

Agreement theorems are no-go results about rational disagreement: if two agents start from a common prior and their posterior beliefs are common knowledge, they cannot assign different probabilities to the same event. Standard treatments of the result have the agents reason about an underlying state of the world, which has lead some to ask whether the result can extend to quantum or postquantum phenomena, where such a description may no longer be appropriate. We derive an operational version of Aumann's agreement theorem without assuming an objective state of the world and instead focusing only on what is observed. This allows us to establish the theorem's validity in quantum theory and even in situations with indefinite causal order or involving hypothetical postquantum phenomena. We comment on seemingly contradictory results in the literature and point to the one place where the theorem might fail: Wigner's friend-type situations.

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

Abstract

Agreement theorems are no-go results about rational disagreement: if two agents start from a common prior and their posterior beliefs are common knowledge, they cannot assign different probabilities to the same event. Standard treatments of the result have the agents reason about an underlying state of the world, which has lead some to ask whether the result can extend to quantum or postquantum phenomena, where such a description may no longer be appropriate. We derive an operational version of Aumann's agreement theorem without assuming an objective state of the world and instead focusing only on what is observed. This allows us to establish the theorem's validity in quantum theory and even in situations with indefinite causal order or involving hypothetical postquantum phenomena. We comment on seemingly contradictory results in the literature and point to the one place where the theorem might fail: Wigner's friend-type situations.
Paper Structure (5 sections, 2 theorems, 17 equations, 1 figure)

This paper contains 5 sections, 2 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical Aumann's theorem
  3. Main result
  4. Quantum theory and beyond
  5. Discussion

Key Result

Theorem 1

If Alice's posterior is $q_A$ and Bob's posterior is $q_B$ and this is common knowledge at the true state of the world $\omega^* \in \Omega$, then $q_A=q_B$. Equivalently, two Bayesian agents with a common prior cannot have different posterior probabilities for the same event if those posteriors are

Figures (1)

  • Figure 1: Left: In the classical agreement theorem, Alice and Bob share a common probability measure $p$ on the state space $\Omega$, and their measurements are described by partitions of $\Omega$; outcomes $i$ and $j$ occur when ${\omega \in \Pi_i^{(A)} \cap \Pi_j^{(B)}}$, and the event $E$ is a subset of $\Omega$. Right: The operational agreement theorem makes no reference to an underlying state of the world and is formulated directly in terms of measurement outcomes.

Theorems & Definitions (4)

  • Definition 1: Common knowledge of posteriors
  • Theorem 1: Aumann's Agreement Theorem
  • Theorem 2: Operational agreement theorem
  • proof