A no-go result on common cause approaches via Hardy relations

Abstract

According to a conventional view, there exists no common cause model of quantum correlations satisfying locality requirements. Indeed, Bell's inequality is derived from some locality requirements and the assumption that the common cause exists, and the violation of the inequality has been experimentally verified. On the other hand, some researchers argued that in the derivation of the inequality, the existence of a common common-cause for multiple correlations is implicitly assumed and that the assumption is unreasonably strong. According to their idea, what is necessary for explaining the quantum correlation is a common cause for each correlation. However, Graβhoff et al. showed that when there are three pairs of perfectly correlated events and a common cause of each correlation exist, we cannot construct a common cause model that is consistent with quantum mechanical prediction and also meets several locality requirements. In this paper, first, as a consequence of the fact shown by Graβhoff et al., we will confirm that there exists no local common cause model when a two-particle system is in any maximally entangled state. After that, based on Hardy's famous argument, we will prove that there exists no local common cause model when a two-particle system is in any non-maximally entangled state. Therefore, it will be concluded that for any entangled state, there exists no local common cause model. It will be revealed that the non-existence of a common cause model satisfying locality is not limited to a particular state like the singlet state.