Characterizing common cause closedness of quantum probability theories

• Common cause closedness is characterized in the category of von Neumann algebras.
• Relation of common cause closedness and the Common Cause Principle is discussed.
• Open problems related to common cause closedness are formulated.

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 Gyenis and Rédei (2014). The result is discussed from the perspective of status of the Common Cause Principle. Open problems on common cause closedness of general probability spaces (L,ϕ)(L,ϕ) are formulated, where LL is an orthomodular bounded lattice and ϕ   is a probability measure on LL.