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The Quantum Agreement Theorem

María García Díaz, Adam Brandenburger, Giannicola Scarpa

TL;DR

This work formulates a quantum analog of the classical Agreement Theorem by introducing a certainty-based epistemic framework for two agents sharing a quantum system. It shows that in the commuting regime, common certainty of probability estimates for a shared property $E$ implies exact agreement, while in the non-commuting regime a distinct phenomenon, common certainty of disagreement (CCD), can occur; agreement is recovered if measurement outcomes are recorded in a classical register. The authors provide a $0-1$ impossibility bound that rules out extreme epistemic divergence, and they derive quantitative robustness bounds on disagreements under state perturbations and approximate certainty. Overall, the paper clarifies how intersubjectivity in quantum mechanics can be rigorously analyzed using epistemic logic, linking classical results to quantum settings and offering tools relevant to QBism, relational QM, and quantum information theory.

Abstract

We formulate and prove an Agreement Theorem for quantum mechanics (QM), describing when two agents, represented by separate laboratories, can or cannot maintain differing probability estimates of a shared quantum property of interest. Building on the classical framework (Aumann, 1976), we define the modality of "common certainty" through a hierarchy of certainty operators acting on each agent's Hilbert space. In the commuting case -- when all measurements and event projectors commute -- common certainty leads to equality of the agents' conditional probabilities, recovering a QM analog of the classical theorem. By contrast, when non-commuting operators are allowed, the certainty recursion can stabilize with different probabilities. This yields common certainty of disagreement (CCD) as a distinctive QM phenomenon. Agreement is restored once measurement outcomes are recorded in a classical register. The classical Agreement Theorem can therefore be seen as emergent from the quantum world via recording. We establish an impossibility result stating that QM forbids a scenario where one agent is certain that a property of interest occurs, and is also certain that the other agent is certain that the property does not occur. In this sense, QM admits non-classical disagreement, but disagreement is still bounded in a disciplined way. We argue that our analysis offers a rigorous approach to the longstanding issue of how to understand intersubjectivity across agents in QM.

The Quantum Agreement Theorem

TL;DR

This work formulates a quantum analog of the classical Agreement Theorem by introducing a certainty-based epistemic framework for two agents sharing a quantum system. It shows that in the commuting regime, common certainty of probability estimates for a shared property implies exact agreement, while in the non-commuting regime a distinct phenomenon, common certainty of disagreement (CCD), can occur; agreement is recovered if measurement outcomes are recorded in a classical register. The authors provide a impossibility bound that rules out extreme epistemic divergence, and they derive quantitative robustness bounds on disagreements under state perturbations and approximate certainty. Overall, the paper clarifies how intersubjectivity in quantum mechanics can be rigorously analyzed using epistemic logic, linking classical results to quantum settings and offering tools relevant to QBism, relational QM, and quantum information theory.

Abstract

We formulate and prove an Agreement Theorem for quantum mechanics (QM), describing when two agents, represented by separate laboratories, can or cannot maintain differing probability estimates of a shared quantum property of interest. Building on the classical framework (Aumann, 1976), we define the modality of "common certainty" through a hierarchy of certainty operators acting on each agent's Hilbert space. In the commuting case -- when all measurements and event projectors commute -- common certainty leads to equality of the agents' conditional probabilities, recovering a QM analog of the classical theorem. By contrast, when non-commuting operators are allowed, the certainty recursion can stabilize with different probabilities. This yields common certainty of disagreement (CCD) as a distinctive QM phenomenon. Agreement is restored once measurement outcomes are recorded in a classical register. The classical Agreement Theorem can therefore be seen as emergent from the quantum world via recording. We establish an impossibility result stating that QM forbids a scenario where one agent is certain that a property of interest occurs, and is also certain that the other agent is certain that the property does not occur. In this sense, QM admits non-classical disagreement, but disagreement is still bounded in a disciplined way. We argue that our analysis offers a rigorous approach to the longstanding issue of how to understand intersubjectivity across agents in QM.

Paper Structure

This paper contains 5 sections, 8 theorems, 97 equations, 5 figures, 3 tables.

Table of Contents

  1. Classical Agreement Theorem with the Certainty Modality
  2. Proof of of the Commutative Agreement Theorem
  3. Additional Scenarios
  4. Proof of the 0 -1 Impossibility Theorem
  5. Proofs of Bounds on Disagreement

Key Result

Theorem S.1

If it is common certainty at state $\rho$ that Alice assigns $q_A$ to $E$ and Bob assigns $q_B$ to $E$, and $p(C_\infty) > 0$, then $q_A = q_B$.

Figures (5)

  • Figure 1: Three-laboratory scenario
  • Figure 2: Information pooling
  • Figure 3: Two-laboratory scenario
  • Figure 4: Single-laboratory scenario
  • Figure 5: Implication structure

Theorems & Definitions (16)

  • Theorem S.1
  • proof
  • Lemma S.1
  • proof
  • Lemma S.2
  • proof
  • Lemma S.3
  • proof
  • Lemma S.4
  • proof
  • ...and 6 more