A mathematical rule against disagreement survives beyond quantum mechanics

Two people can examine the same evidence and reach different conclusions. However, if they share the same starting assumptions and know what the other believes, a theorem developed nearly five decades ago says their disagreement cannot survive under certain conditions.

Now, two Chapman University researchers have shown that this principle works beyond classical probability, including in a framework that combines quantum theory with a classical model of knowledge and a wide range of generalized probability theories.

“We argue that agreeing to disagree is impossible in quantum theory. Secondly, and based on the quantum argument, we show that agreeing to disagree is also forbidden in any generalised probability theory,” the researchers note in their study.

This work suggests that the limits of disagreement may come less from the nature of reality and more from how information is updated.

The problem with taking agreement beyond classical physics

In 1976, economist Robert Aumann established the Agreement Theorem. It states that when rational agents begin with the same prior beliefs and their updated beliefs become common knowledge, they cannot continue assigning different probabilities to the same event.

Common knowledge means more than everyone knowing something. Each person knows it, knows that the others know it, and so on indefinitely.

The theorem originally relied on classical probability. This leads to an important question: Could quantum mechanics, with its unusual rules for describing physical systems, allow people to agree to disagree?

Previous research explored extensions involving quantum scenarios and non-signaling systems. The authors of the current study took a different route, examining whether the theorem could survive when the mathematical description of probability changed.

A mathematical rule that survives quantum theory

The researchers first developed a quantum version of the Agreement theorem using a classical, set-theoretical framework for representing knowledge alongside quantum mathematical objects called density operator-valued measures (DOVMs).

A DOVM allows researchers to describe information about events using quantum states rather than ordinary numerical probabilities. The researchers then defined a conditional quantum state, representing information available after restricting attention to a particular event.

This was crucial because Aumann’s original proof depends on conditioning—the process of updating a probability after acquiring information.

They showed that when agents’ conditional quantum states become common knowledge, those states must be identical, provided the common-knowledge event has a nonzero quantum measure.

However, this result applies to their particular hybrid framework, not every possible framework for representing quantum knowledge. The researchers then extended their reasoning to generalized probability theories (GPTs), which allow scientists to study probability systems beyond classical and quantum theory.

Their approach relies on two requirements. The first is a probability-like measure that adds consistently across mutually exclusive events. The second is a well-defined method for conditioning, or updating a state based on information.

“In its probabilistic version, the Agreement Theorem is a direct consequence of how we choose to condition upon acquiring new information,” the study authors added.

Using state-valued measures and conditional states, they showed that Aumann’s theorem applies to these systems when the necessary mathematical conditions are satisfied.

What the result does, and does not establish

The study shows that Aumann’s Agreement Theorem is not limited to classical probability or quantum theory. Instead, its validity depends on the mathematical rules governing information updates.

However, the researchers’ framework has limitations. It uses a static model of knowledge and cannot represent certain joint quantum states, including those involving non-Markovian processes and pre- and post-selection scenarios.

The researchers suggest that future work could examine communication between agents and broader versions of the theorem.

For now, their findings establish that changing the rules of probability does not necessarily remove the mathematical conditions that prevent rational agents from agreeing to disagree.

The study is published in the journal Quantum.