Characterizing common cause closedness of quantum probability theories

Miklós Rédei
London School of Economics
We prove new results on common cause closedness of quantum probability spaces, where by a quantum probability space is meant the projection lattice of a non-commutative von Neumann algebra together with a countably additive probability measure on the lattice. Common cause closedness is the feature that for every correlation between a pair of commuting projections there exists in the lattice a third projection commuting with both of the correlated projections and which is a Reichenbachian common cause of the correlation. The main result we prove is that a quantum probability space is common cause closed if and only if it has at most one measure theoretic atom. This result improves earlier ones published in [1]. The result is discussed from the perspective of status of the Common Cause Principle. Open problems on common cause closedness of general probability spaces are formulated, where L is an orthomodular bounded lattice and ϕ is a probability measure on L.
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DOI 10.1016/j.shpsb.2015.08.003
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Venetian Sea Levels, British Bread Prices, and the Principle of the Common Cause.Elliott Sober - 2001 - British Journal for the Philosophy of Science 52 (2):331-346.
Quantum Probability Theory.Miklós Rédei & Stephen Jeffrey Summers - 2007 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 38 (2):390-417.

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