Einstein-Podolsky-Rosen (EPR) correlations Explaining the correlations predicted by quantum mechanics in the case of joint quantum systems of the EPR type has proved to be a major challenge for the Common Cause Principle. The EPR correlations have been confirmed by a large number of sophisticated experiments carried out since the early 1980s, and, since quantum theory does not give us any hints as to where to look for possible common causes that could explain the correlations in question, the suspicion arose that common causes of EPR correlations might not exist at all. If this were indeed the case, then the EPR correlations would constitute strong empirical evidence against Reichenbach's Common Cause Principle. But how could this be possible in view of the results presented in the previous chapters? As we have seen (Chapter 3), every common cause incomplete probability space is common cause extendible (even strongly common cause extendible with respect to any finite set of correlations; Proposition 3.9); therefore every correlation can, in principle, be explained by a (possibly hidden) common cause, including the (finite number of) EPR correlations. Could the EPR correlations (not) be explained by referring to such hidden common causes? To answer this question, one has to realize first that it is not quite obvious how to link the Common Cause Principle to quantum mechanics because Reichenbach's notion of common cause was defined in terms of classical probability theory (Chapter 2), not in terms of quantum probability theory.
Einstein-Podolsky-Rosen (EPR) correlations Explaining the correlations predicted by quantum mechanics in the case of joint quantum systems of the EPR type has proved to be a major challenge for the Common Cause Principle. The EPR correlations have been confirmed by a large number of sophisticated experiments carried out since the early 1980s, and, since quantum theory does not give us any hints as to where to look for possible common causes that could explain the correlations in question, the suspicion arose that common causes of EPR correlations might not exist at all. If this were indeed the case, then the EPR correlations would constitute strong empirical evidence against Reichenbach's Common Cause Principle. But how could this be possible in view of the results presented in the previous chapters? As we have seen (Chapter 3), every common cause incomplete probability space is common cause extendible (even strongly common cause extendible with respect to any finite set of correlations; Proposition 3.9); therefore every correlation can, in principle, be explained by a (possibly hidden) common cause, including the (finite number of) EPR correlations. Could the EPR correlations (not) be explained by referring to such hidden common causes? To answer this question, one has to realize first that it is not quite obvious how to link the Common Cause Principle to quantum mechanics because Reichenbach's notion of common cause was defined in terms of classical probability theory (Chapter 2), not in terms of quantum probability theory.
Reichenbach's Common Cause Principle is the claim that if two events are correlated, then either there is a causal connection between the correlated events that is responsible for the correlation or there is a third event, a so called (Reichenbachian) common cause, which brings about the correlation. The paper reviews some results concerning Reichenbach's notion of common cause, results that are directly relevant to the problem of how one can falsifY Reichenbach's Common Cause Principle. Special emphasis will be put on the question of whether EPR-type correlations can have an explanation in terms of Reichenbachian common causes. Most of the results to be recalled indicate that falsifYing Reichenbach's Common Cause Principle is much more tricky than one may have thought and that, contrary to some claims in the literature, there is no conclusive proof yet that the EPR correlations predicted by ordinary, non-relativistic quantum mechanics cannot have a common cause; furthermore, recent results about possible Reichenbachian common causes of correlations predicted by quantum field states between spacelike separated local observable algebras in algebraic quantum field theory strongly indicate that there may very well exist Reichenbachian common causes of superluminal correlations predicted by quantum field theory.
Common causes and common common causes Proposition 3.9 tells us that every common cause incomplete classical probability space can be strongly common cause extended with respect to any (finite) set of correlations; Proposition 4.19 states that every classical probability space is even common cause completable. Note that what these propositions say is not that for a set of correlations between ( A i , B i )( i = 1, 2 … n ) there exists a single, common common cause C in the extension (or completion) for the whole set of correlations; in fact, the common causes C i constructed explicitly in the proof of Proposition 3.9 are all different: C i ≠ C j ( i ≠ j ). This observation leads to the following question. Let ( A i , B i )( i = 1, … n )be a finite set of pairs of events in( X, S, p )that are correlated [ Corr p ( A i , B i ) ＞ 0 for every i ]. We say that C is a common common cause of these correlations if C is a Reichenbachian common cause of the correlated pair ( A i , B i )for every i . Does every set of correlations in a classical probability space have a common common cause? In view of the generality of this question one may surmise that the answer to it is “yes”; that is to say, one may conjecture that given any two correlations there can always exist a Reichenbachian common cause which is a common cause for both correlations, since, one may reason, we just have to refine our picture of the world by adding more and more events to the original event structure in a consistent manner, and finally we shall find a single common cause that explains both correlations.
Common cause partitions Confronted with a common cause incomplete probability space ( X, S, p )in which a direct causal influence between the correlated events is excluded, one can have in principle two strategies aiming at saving the Common Cause Principle: one may try to argue that S is not “rich enough” to contain a common cause, but there might exist a larger probability space ( X ′, S ′, p ′) that already contains a common cause of the correlation. As we have seen in Chapter 3 this strategy always works in the sense that it is always possible to enlarge ( X, S, p ) in such a way that the enlarged probability space already contains an event C that satisfies the Reichenbachian conditions. Another natural idea is to suspect that the correlation between A and B is not due to a single factor but may be the cumulative result of a (possibly large) number of different “partial common causes,” none of which can in and by itself yield a complete common-cause-type explanation of the correlation, all of which, taken together, can however account for the entire correlation. In this chapter we elaborate this idea by formulating precisely a notion of the Reichenbachian Common Cause System (RCCS) and prove propositions on the existence and features of such systems. As we have seen in Chapter 2, if the events A, B, C satisfy the Reichenbachian conditions (2.5)–(2.8) then there is a positive correlation between A and B (Proposition 2.5).
From a phenomenological and experimental viewpoint, quantum mechanics is remarkably successful in explaining effects and processes in the microscopic world. Recently, however, a growing body of literature has revisited the theory's own formal foundations, specifically addressing issues of self-consistency and completeness. As a major historical example of these foundational challenges, the Einstein-Podolsky-Rosen (EPR) argument famously claimed that the theory is incomplete. While many valid criticisms and refutations of the EPR logic exist, they generally rely on a combination of technical results and conceptual objections to EPR's interpretive assumptions, thus transcending the plain quantum framework itself. Motivated by these trends of examining the theory's structural limits, here we revisit the EPR argument. Focusing on elements essential to the EPR reasoning, we first analyze general quantum correlations for EPR states: (i) the fact that their observables are always associated with non-commuting operators, and (ii) the nature of the correlated information obtained through measurement. From these two points alone, and relying strictly on the core rules of quantum mechanics, we demonstrate how to overturn the alleged EPR incompleteness without invoking any extraneous propositions. Consequently, we show that the standard formalism alone suffices to resolve such type of skepticism regarding the theory's physical reach. This offers, at least in a paradigmatic instance, a powerful indication of a structurally well-founded, self-consistent quantum theory.
It is shown that, given any finite set of pairs of random events in a Boolean algebra which are correlated with respect to a fixed probability measure on the algebra, the algebra can be extended in such a way that the extension contains events that can be regarded as common causes of the correlations in the sense of Reichenbach's definition of common cause. It is shown, further, that, given any quantum probability space and any set of commuting events in it which are correlated with respect to a fixed quantum state, the quantum probability space can be extended in such a way that the extension contains common causes of all the selected correlations, where common cause is again taken in the sense of Reichenbachs definition. It is argued that these results very strongly restrict the possible ways of disproving Reichenbach's Common Cause Principle.
No correlation without causation. This is, in its most compact and general formulation, the essence of what has become Reichenbach's Common Cause Principle . More explicitly, the Common Cause Principle says that every correlation is either due to a direct causal effect linking the correlated entities, or is brought about by a third factor, a so-called Reichenbachian common cause that stands in a well-defined probabilistic relation to the correlated events, a relation that explains the correlation in the sense of entailing it. The Common Cause Principle is a nontrivial metaphysical claim about the causal structure of the World and entails that all correlations can (hence, should) be explained causally either by pointing at a causal connection between the correlated entities or by displaying a common cause of the correlation. Thus, the Common Cause Principle licenses one to infer causal connections from probabilistic relations; at the same time the principle does not address whether the causal connection holds between the correlated entities or between the common cause and the elements in the correlation. While the technically explicit notion of common cause of a probabilistic correlation within the framework of classical Kolmogorovian probability theory is due to Reichenbach (1956), the Common Cause Principle was articulated explicitly only later, especially in the works by W. Salmon (see the “Notes and bibliographic remarks” to Chapter 2). The chief aim of this book is to investigate the Common Cause Principle; in particular, the problem of to what extent the Common Cause Principle can explain probabilistic correlations.