Quantum Bayes' Rule and Petz Transpose Map from the Minimum Change Principle

Area of Science:

• Quantum Information Theory
• Foundations of Probability

Background:

• Bayes' rule updates beliefs with new evidence, derived from a minimum change principle.
• Classical Bayes' rule minimizes changes between joint input-output distributions and their marginals.

Purpose of the Study:

• Introduce a quantum analog of the minimum change principle.
• Derive a quantum Bayes' rule by minimizing changes between quantum input-output processes.

Main Methods:

• Formulated a quantum minimum change principle.
• Minimized the change between quantum input-output processes, not just their marginals.
• Maximized fidelity for a unique solution.

Main Results:

• Derived a quantum Bayes' rule based on the quantum minimum change principle.
• The quantum Bayes' rule recovers the Petz transpose map under fidelity maximization.

Conclusions:

• The quantum minimum change principle provides a novel foundation for quantum probability updating.
• This work establishes a quantum analog to classical Bayesian inference with potential applications in quantum information processing.