The Quantum Agreement Theorem

TL;DR

This paper develops a quantum version of the Agreement Theorem, analyzing when agents in quantum mechanics can or cannot agree on probabilities of shared properties, revealing conditions for agreement and disagreement.

Contribution

It introduces a quantum Agreement Theorem framework, extending classical agreement concepts to quantum settings, including non-commuting measurements and classical recording effects.

Findings

In commuting cases, agents' probabilities agree, mirroring classical results.

Non-commuting measurements can lead to stable disagreement, termed CCD.

Agreement is restored when measurement outcomes are recorded classically.

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. We show that agreement will nevertheless re-emerge if measurement outcomes are recorded in a classical register. We also 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.