Quantum Bayes' rule and Petz transpose map from the minimum change principle

TL;DR

This paper introduces a quantum version of the minimum change principle to derive a quantum Bayes' rule, which generalizes classical Bayesian updating and relates to the Petz transpose map, providing a new perspective on quantum belief updates.

Contribution

It develops a quantum analog of the minimum change principle and derives a quantum Bayes' rule that connects to the Petz transpose map, expanding quantum Bayesian inference methods.

Findings

Quantum Bayes' rule derived from minimum change principle.

When maximizing fidelity, the rule recovers the Petz transpose map.

Provides a new framework for quantum belief updating.

Abstract

Bayes' rule, which is routinely used to update beliefs based on new evidence, can be derived from a principle of minimum change. This principle states that updated beliefs must be consistent with new data, while deviating minimally from the prior belief. Here, we introduce a quantum analog of the minimum change principle and use it to derive a quantum Bayes' rule by minimizing the change between two quantum input-output processes, not just their marginals. This is analogous to the classical case, where Bayes' rule is obtained by minimizing several distances between the joint input-output distributions. When the change maximizes the fidelity, the quantum minimum change principle has a unique solution, and the resulting quantum Bayes' rule recovers the Petz transpose map in many cases.