On modifications of Reichenbach's principle of common cause in light of Bell's theorem

Bell's 1964 theorem causes a severe problem for the notion that correlations require explanation, encapsulated in Reichenbach's Principle of Common Cause. Despite being a hallmark of scientific thought, dropping the principle has been widely regarded as a much less bitter medicine than the perceived alternative---dropping relativistic causality. Recently, however, some authors have proposed that modified forms of Reichenbach's principle could be maintained even with relativistic causality. Here we break down Reichenbach's principle into two independent assumptions---the Principle of Common Cause proper, and Factorisation of Probabilities. We show how Bell's theorem can be derived from these two assumptions plus Relativistic Causality and the Law of Total Probability for actual events, and we review proposals to drop each of these assumptions in light of the theorem. In particular, we show that the non-commutative common causes of Hofer-Szabo and Vecsernyes fail to have an analogue of the notion that the common causes can explain the observed correlations. Moreover, we show that their definition can be satisfied trivially by any quantum product state for any quantum correlations. We also discuss how the conditional states approach of Leifer and Spekkens fares in this regard.