Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference

TL;DR

This paper develops a unified formalism of quantum conditional states that treats quantum experiments and inferences in a causally neutral way, generalizing classical Bayesian inference to quantum theory.

Contribution

It introduces quantum conditional states, unifies various quantum concepts under belief propagation, and reformulates quantum inference using a generalized Bayesian framework.

Findings

Quantum conditional states unify channels, states, and measurements.

Belief propagation describes quantum state updates and inference.

Quantum Bayes' theorem enables causal-neutral quantum reasoning.

Abstract

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. Whereas the manner in which inferences are made in classical probability theory is independent of the causal relation that holds between the conditioned variable and the conditioning variable, in the conventional quantum formalism, there is a significant difference between how one treats experiments involving two systems at a single time and those involving a single system at two times. In this article, we develop the formalism of quantum conditional states, which provides a unified description of these two sorts of experiment. In addition, concepts that are distinct in the conventional formalism become unified: channels, sets of states, and positive operator valued measures are all seen to be instances of conditional states; the action of a channel on a state, ensemble averaging, the Born rule, the composition of channels, and nonselective state-update rules are all seen to be instances of belief propagation. Using a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning, we also show that the remote steering of quantum states can be described within our formalism as a mere updating of beliefs about one system given new information about another, and retrodictive inferences can be expressed using the same belief propagation rule as is used for predictive inferences. Finally, we show that previous arguments for interpreting the projection postulate as a quantum generalization of Bayesian conditioning are based on a misleading analogy and that it is best understood as a combination of belief propagation (corresponding to the nonselective state-update map) and conditioning on the measurement outcome.