Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference

By Matthew S. Leifer, Robert W. Spekkens. 2011

325 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

  1. Roger Colbeck
  2. Robert W. Spekkens
  3. Terry Rudolph
  4. Matthew F Pusey
  5. Ciarán M. Lee
  6. Raymond Lal
  7. Joe Henson
  8. Matthew S. Leifer

In Atomic and Molecular Physics, and Optics

  1. Roger Colbeck
  2. Matthew S. Leifer
  3. Terry Rudolph
  4. Klaas Landsman
  5. Robert W. Spekkens
  6. Ciarán M. Lee
  7. Matthew F Pusey
  8. Jonathan Barrett

In Statistical and Nonlinear Physics

  1. Matthew S. Leifer
  2. Roger Colbeck
  3. Klaas Landsman
  4. Terry Rudolph
  5. Robert W. Spekkens
  6. Renato Renner

Citations in the corpus