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Slovak Journal of Civil Engineering

A note to a common cause in quantum logic

Ahmed Al-Adilee

Published Dec 1, 2010• 1 citation
Publisher

Abstract

A note to a common cause in quantum logic We study the notion of a common cause in quantum logics as a type of causal relations (see [13]). We review maps like conditional state, s-map as basic tools of our constructions. Finally, we show some properties of such notions in quantum logics.
DOI
10.2478/v10189-010-0019-z

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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 common cause. The principle is of central importance in the philosophy of science, especially in causal explanation, causal modeling and in the foundations of quantum physics. Written for philosophers of science, physicists and statisticians, this book contributes to the debate over the validity of the common cause principle, by proving results that bring to the surface the nature of explanation by common causes. It provides a technical and mathematically rigorous examination of the notion of common cause, providing an analysis not only in terms of classical probability measure spaces, which is typical in the available literature, but in quantum probability theory as well. The authors provide numerous open problems to further the debate and encourage future research in this field.
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.
It has long been recognized as a difficult problem to determine whether the observed statistical correlations between two classical variables arise from causality or from common causes. Recent research has shown that, in a quantum theoretical framework, the mechanisms of entanglement and quantum coherence provide advantages in tackling this problem. In some particular cases, quantum common causes and quantum causality can be effectively distinguished by using observations alone. However, these solutions do not apply to all cases. There still exist a large class of cases in which quantum common causes and quantum causality cannot be distinguished. In this paper, along the line of considering unitary transformation as causality in the quantum world, we formally show that quantum common causes and quantum causality are universally separable. Based on the analysis, we further provide a general method to discriminate the two.
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.
Understanding the causal influences that hold among parts of a system is critical both to explaining that system's natural behaviour and to controlling it through targeted interventions. In a quantum world, understanding causal relations is equally important, but the set of possibilities is far richer. The two basic ways in which a pair of time-ordered quantum systems may be causally related are by a cause-effect mechanism or by a common-cause acting on both. Here we show a coherent mixture of these two possibilities. We realize this nonclassical causal relation in a quantum optics experiment and derive a set of criteria for witnessing the coherence based on a quantum version of Berkson's effect, whereby two independent causes can become correlated on observation of their common effect. The interplay of causality and quantum theory lies at the heart of challenging foundational puzzles, including Bell's theorem and the search for quantum gravity.