Causal Completeness in General Probability Theories

Michał Marczyk, Leszek Wroński

Jonathan Steinberg

Matthew Leifer, Robert W. Spekkens

Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Classical probability theory is independent of causal structure, whereas the conventional quantum formalism requires causal structure to be fixed in advance. In this paper, we develop the formalism of quantum conditional states, which unifies the description of experiments involving two systems at a single time with the description of those involving a single system at two times. The analogies between quantum theory and classical probability theory are expressed succinctly within the formalism and it unifies the mathematical description of distinct concepts, such as ensemble preparation procedures, measurements, and quantum dynamics. We introduce a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning. Conditioning a quantum state on a classical variable is the correct rule for updating quantum states in light of classical data, regardless of the causal relationship between the classical variable and the quantum degrees of freedom, but it does not include the projection postulate as a special case. We show that previous arguments that projection is the quantum generalization of conditioning are based on misleading analogies. Since our formalism is causally neutral, conditioning provides a unification of the predictive and retrodictive formalisms for prepare-and-measure experiments and leads to an elegant derivation of the set of states that a system can be steered to by making measurements on a remote system.

Michael J. W. Hall

Michał Marczyk, Leszek Wroński

We give a few results concerning the notions of causal completability and causal closedness of classical probability spaces (Hofer-Szabó et al. [1999]; Gyenis and Rédei [2004]). We prove that (i) any classical probability space has a causally closed extension; (ii) any finite classical probability space with positive rational probabilities on the atoms of the event algebra can be extended to a causally up-to-three-closed finite space; and (iii) any classical probability space can be extended to a space in which all correlations between events that are logically independent modulo measure zero event have a countably infinite common-cause system. Collectively, these results show that it is surprisingly easy to find Reichenbach-style ‘explanations' for correlations, underlining doubts as to whether this approach can yield a philosophically relevant account of causality. 1 Introduction2 Basic Definitions and Results in the Literature3 Causal Completability the Easy Way: ‘Splitting the Atom’4 Causal Completability of Classical Probability Spaces: The General Case5 Infinite Statistical Common-Cause Systems for Arbitrary Pairs6 Conclusion Appendix A

Miklós Rédei, Balázs Gyenis

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