Formulating Quantum Theory as a Causally Neutral Theory of Bayesian Inference
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.