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Formulating Quantum Theory as a Causally Neutral Theory of Bayesian Inference


Quantum theory can be thought of as a noncommutative generalization of Bayesian probability theory, but for the analogy to be convincing, it should be possible to describe inferences among quantum systems in a manner that is independent of the causal relationship between those systems.  In particular, it should be possible to unify the treatment of two kinds of inferences: (i) from beliefs about one system to beliefs about another, for instance, in the Einstein-Podolsky-Rosen or "quantum steering" phenomenon, and (ii) from beliefs about a system at one time to beliefs about that same system at another time, for instance, in predictions or retrodictions about a system undergoing dynamical evolution.  I will present a formalism that achieves such a unification by making use of "conditional quantum states", a noncommutative generalization of conditional probabilities.  Elements of the conventional formalism, such as sets of states, positive operator valued measures, quantum operations and quantum instruments become special cases of conditional states, and familiar formulas, such as Born's rule, the expression for the ensemble average, the rule for dynamical evolution, and the nonselective measurement-update rule become special cases of belief propagation.  (Joint work with Matthew Leifer)