The quantum agreement theorem

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Abstract
Intersubjective consistency asks when different observers of a physical system, reasoning from a shared theory, will agree on their probability assessments. Classical probability theory provides a benchmark in the form of the Agreement Theorem [R. J. Aumann, Ann. Stat. 4 , 1236–1239 (1976)]. We study the quantum mechanics (QM) analog to the classical Agreement Theorem in a finite-dimensional tripartite setting in which two observers obtain information via local projective measurements. We represent their mutual awareness by common certainty, which is formally an infinite hierarchy of certainty operators concerning the probability of an event of interest. We first prove that if the observers’ measurements commute with one another and with the property of interest, common certainty forces their probability assessments to coincide, thereby recovering a QM analog to the classical Agreement Theorem. We then construct a one-parameter family of qutrit–qubit–qubit models in which one observer’s measurement does not commute with the property and common certainty of disagreement occurs—a distinctively QM phenomenon. The mechanism is coherence between branches distinguished by one but not the other observer’s measurement. When measurement outcomes are stored in a classical register, the resulting dephasing removes this coherence and restores agreement. Finally, we prove that QM does not permit the extreme 0 - 1 configuration in which Alice is certain of a property and is also certain that Bob is certain of its negation. These results aim to turn debate in QM about observer-dependent facts into precise mathematical conditions for when intersubjective agreement or disagreement in QM can be sustained.

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