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

Abstract: Bayes' rule, which is routinely used to update beliefs based on new evidence, arises from a principle of minimal 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 minimal 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 the change between the joint input-output distributions. When the change is quantified by the fidelity, the quantum minimal change principle has a unique solution, and the resulting quantum Bayes' rule recovers the Petz transpose map.