# Reichenbachian common cause systems of size 3 in general probability theories

## TL;DR

This paper investigates Reichenbachian common cause systems of size 3 within general probability theories, including quantum theories, showing their existence under certain conditions and exploring their relation to Bell's inequality.

## Contribution

It demonstrates the existence of size 3 Reichenbachian common cause systems in general probability theories for compatible, independent events and analyzes their relation to Bell's inequality.

## Key findings

- No common cause system exists for correlated, non-independent events.
- Size 3 systems exist for compatible, independent events in atomless orthomodular lattices.
- Bell's inequality violation can coexist with Reichenbachian common cause systems.

## Abstract

Reichenbach defined a common cause which explains a correlation between two events if either one does not cause the other. Its intuitive idea is that the statistical ensemble can be divided into two disjoint parts so that the correlation disappears in both of the resulting subensembles if there is no causal connection between these correlated events. These subensembles can be regarded as common causes. Hofer-Szab\'{o} and R\'{e}dei (2004) generalized a Reichenbachian common cause, and called it a Reichenbachian common cause system. In the case of Reichenbachian common cause systems the statistical ensemble is divided more than two, while it is divided into two parts in the case of Reichenbachian common causes. The number of these subensembles is called the size of this system. In the present paper, we examine Reichenbachian common cause systems in general probability theories which include classical probability theories and quantum probability theories. It is shown that there is no Reichenbachian common cause system for any correlation between two events which are not logical independent, and that a general probability theory which is represented by an atomless orthomodular lattice with a faithful $\sigma$-additive probability measure contains Reichenbachian common cause systems of size 3 for any correlation between two compatible and logical independent events. Moreover, we discuss a relation between Bell's inequality and Reichenbachian common cause systems, and point out that this violation of Bell's inequality can be compatible with a Reichenbachian common cause system although it contradicts a `common' common cause system.