Quantum Information Journal Club: ‘The Quantum Agreement Theorem’
April 22 | 11:30 - 12:30
Title: The Quantum Agreement Theorem
Speaker: María García (upm)
Venue&Time: Gray Room 3 / 11:30
Abstract: We formulate and prove an Agreement Theorem for quantummechanics (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.
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.