Characterizing common cause closedness of quantum probability theories

TL;DR

This paper establishes a precise condition under which quantum probability spaces are common cause closed, linking this property to the presence of at most one measure-theoretic atom, thus advancing understanding of the Common Cause Principle.

Contribution

It proves that a quantum probability space is common cause closed if and only if it has at most one measure-theoretic atom, refining previous results in the field.

Findings

Quantum probability space is common cause closed iff it has at most one measure-theoretic atom.

Improves earlier results on common cause closedness in quantum probability spaces.

Discusses implications for the status of the Common Cause Principle.

Abstract

We prove new results on common cause closedness of quantum probability spaces, where by a quantum probability space is meant the projection lattice of a non-commutative von Neumann algebra together with a countably additive probability measure on the lattice. Common cause closedness is the feature that for every correlation between a pair of commuting projections there exists in the lattice a third projection commuting with both of the correlated projections and which is a Reichenbachian common cause of the correlation. The main result we prove is that a quantum probability space is common cause closed if and only if it has at most one measure theoretic atom. This result improves earlier ones published in Z. GyenisZ and M. Redei Erkenntnis 79 (2014) 435-451. The result is discussed from the perspective of status of the Common Cause Principle. Open problems on common cause closedness of general probability spaces $(\mathcal{L},\phi)$ are formulated, where $\mathcal{L}$ is an orthomodular bounded lattice and $\phi$ is a probability measure on $\mathcal{L}$.