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arXiv:1503.04410 · Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

Characterizing common cause closedness of quantum probability theories

Yuichiro Kitajima, Miklós Rédei

Published Sep 8, 2015• 4 citations
arXivHTMLPDF
DOI
10.1016/j.shpsb.2015.08.003

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In recent years part of the literature on probabilistic causality concerned notions stemming from Reichenbach’s idea of explaining correlations between not directly causally related events by referring to their common causes. A few related notions have been introduced, e.g. that of a “common cause system” (Hofer-Szabó and Rédei in Int J Theor Phys 43(7/8):1819–1826, 2004 ) and “causal (N-)closedness” of probability spaces (Gyenis and Rédei in Found Phys 34(9):1284–1303, 2004 ; Hofer-Szabó and Rédei in Found Phys 36(5):745–756, 2006 ). In this paper we introduce a new and natural notion similar to causal closedness and prove a number of theorems which can be seen as extensions of earlier results from the literature. Most notably we prove that a finite probability space is causally closed in our sense iff its measure is uniform. We also present a generalisation of this result to a class of non-classical probability spaces.

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Causal closedness and common cause closedness Assuming that Reichenbach's Common Cause Principle is valid, one is led to the question of whether our probabilistic theories predicting probabilistic correlations can be causally rich enough to also contain the causes of all the correlations they predict. The aim of this chapter is to formulate precisely and investigate this question. According to the Common Cause Principle, causal richness of a theory T would manifest in T 's being causally closed (complete) in the sense of being capable of explaining the correlations by containing a common cause of every correlation between causally independent events A, B . This feature of a theory is formulated in the next two definitions of causal closedness. In both definitions ( X, S, p )is a probability space and R ind is a two-place causal independence relation that is assumed to have been defined between elements of S . We treat the relation R ind as a variable in the problem of causal closedness; hence, at this point we leave open what properties R ind should have to be acceptable as a causal independence relation – later we will return to the issue of how to specify it. Definition 4.1 The probability space ( X, S, p ) is called causally closed with respect to R ind , if for every correlation Corr p ( A, B ) ＞ 0 with A ϵ S and B ϵ S such that R ind ( A, B ) holds, there exists a common cause C of some type in S .
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
In recent years part of the literature on probabilistic causality concerned notions stemming from Reichenbach’s idea of explaining correlations between not directly causally related events by referring to their common causes. A few related notions have been introduced, e.g. that of a “common cause system” (Hofer-Szabó and Rédei in Int J Theor Phys 43(7/8):1819–1826, 2004 ) and “causal (N-)closedness” of probability spaces (Gyenis and Rédei in Found Phys 34(9):1284–1303, 2004 ; Hofer-Szabó and Rédei in Found Phys 36(5):745–756, 2006 ). In this paper we introduce a new and natural notion similar to causal closedness and prove a number of theorems which can be seen as extensions of earlier results from the literature. Most notably we prove that a finite probability space is causally closed in our sense iff its measure is uniform. We also present a generalisation of this result to a class of non-classical probability spaces.