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.
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.
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.
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.
The aim of this paper is to give a review of the known results and open problems concerning causal completeness (closedness) of classical and non-classical (quantum) probability spaces. Causal closedness of a probabilistic theory means that the theory is causally rich enough to be able to explain causally all the correlations it predicts. It is natural to ask whether probabilistic theories are causally closed if one assumes that Reichenbach’s Common Cause Principle holds: this principle states that if two events A and B are probabilistically correlated then either the correlation is due to a causal interaction between A and B, or, if A and B are causally independent, Rind(A,B), then there exists a third event C, a so-called common cause that explains the correlation by being related to A and B probabilistically in a specific way (see Definition 1). Causal closedness of a probabilistic theory is intended to express that the theory complies with the Common Cause Principle; accordingly, and more precisely, a probability theory is defined to be causally closed with respect to a causal independence relation
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.
Given a probabilistic correlation between two events, this correlation might be explained in terms of a common cause. In [1] Reichenbach defines the notion of common cause and shows that the definition is consistent with the explicable correlation, i.e. if two events have a common cause then they do correlate. In this paper we generalize the notion of common cause to Hilbert lattices in two different ways according to the two different definitions of conditional probability in the quantum case, and show that Reichenbach's theorem does not hold in either case. There will be given counter-examples when a common cause 'causes' correlation, anticorrelation and independence, respectively.
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.