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.
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.
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.
Reichenbach's principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations.Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize.The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution.In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach's principle would seem to demand.Motivated by this, we introduce a quantum counterpart to the principle.We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input A and outputs B and C is compatible with A being a complete common cause of B and C, then it must factorize in a particular way.Finally, we show how to generalize our quantum version of Reichenbach's principle to a formalism for quantum causal models and provide examples of how the formalism works.
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.
Reichenbach's principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations.Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize.The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution.In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach's principle would seem to demand.Motivated by this, we introduce a quantum counterpart to the principle.We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input A and outputs B and C is compatible with A being a complete common cause of B and C, then it must factorize in a particular way.Finally, we show how to generalize our quantum version of Reichenbach's principle to a formalism for quantum causal models and provide examples of how the formalism works.
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.
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.