A note on Reichenbach’s common cause completeness

Iñaki San Pedro, Mauricio Suárez

Gábor Hofer‐Szabó, Péter Vecsernyés

Jeffrey Bub

Dominika Burešová

Abstract One of the lines of recent quantum logic investigations concern the quantum logics that are endowed with an operation of symmetric difference (with a XOR operation). It is natural to ask whether these logics might be relevant to the study of Reichenbach’s common cause principle (RCCP). The principle states that any two positively correlated events allow for a common cause. In the mathematical formulation, one asks if there are symmetric-difference-closed (non-Boolean) quantum logics that enable a positive answer as regards the RCCP. In this note we show that there are such logics.

Claudio Mazzola

Dominika Burešová, Kamila Houšková, Mirko Navara +3 more

Abstract Reichenbach’s common cause principle (RCCP) is a metaphysical claim about the causal structure of the world. It entails that all correlations can be explained causally either by pointing at the causal connection between the correlated entities or by displaying a common cause of the correlation. We contribute to its mathematical side. Firstly, after adopting the RCCP axioms, we indicate the importance of positive covariance of events in connection with RCCP. In fact, RCCP is meaningful only for positively correlated events. Secondly, we find explicit requirements for a state to have a common cause in a quantum logic. Afterwards, we compare RCCP in the standard (Boolean) case and in the quantum setup that admits new properties of correlation, impossible in the classical case. We show that the notion of maximal correlation differs considerably in these two cases. We answer an open question by providing a counterexample based on a quantum logic given by a free orthomodular lattice. Subsequently, we find a relation between the Darboux property and RCCP. We invent a new technique for obtaining the central result that atomless σ-complete quantum logics are common cause complete. A variety of new examples is presented. Finally, we contribute to a fundamental question whether a common cause incomplete quantum logic can be embedded into one that is common cause complete. We contribute by an advanced construction that allows for a positive answer to this question for quantum logics with finitely many atoms and even for some quantum logics with countably many atoms.