Description
The classical Agreement Theorem (Aumann, 1976) asserts that rational agents with a common prior and common knowledge of their posterior probabilities for an event must assign it the same probability, precluding the possibility of agreeing to disagree. In this work, we extend this principle to quantum mechanics, introducing the Quantum Agreement Theorem for a two-player model of incomplete information with beliefs. Defined over a finite-dimensional Hilbert space with projective measurements and a shared quantum state, our framework formalizes quantum notions of certainty and common certainty among agents Alice and Bob. We prove that if it is common certainty at a measurement outcome that Alice assigns probability qA and Bob assigns qB to an event, E, then qA = qB, provided that specific commutativity conditions hold among the measurements, the event, and the state. This result generalizes the classical theorem to quantum systems, leveraging compatibility to ensure agreement despite the complexities of quantum measurement. We illustrate its applicability through settings involving commuting measurements and entangled states, offering examples such as spin systems and multi-qubit scenarios. The theorem has practical implications for designing quantum communication protocols, where aligned probability estimates enhance security and coordination, and foundational significance for understanding intersubjective agreement and the emergence of classicality in quantum mechanics. Our work bridges classical epistemics and quantum theory, opening new avenues for exploring consensus among quantum observers.
