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

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.

12 pages, 2 figures, updated according to the published version, fixed typos; comments are welcome

References in corpus (25)

• Theoretical framework for quantum networks
• Quantum Circuits Architecture
• Transforming quantum operations: quantum supermaps
• Thermalizing Quantum Machines: Dissipation and Entanglement
• Continuous quantum measurement of a double dot
• Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
• Reversing quantum dynamics with near-optimal quantum and classical fidelity
• Selective quantum evolution of a qubit state due to continuous measurement
• Quantum Bayes rule
• Quantum Graphical Models and Belief Propagation
• Quantum Operation Time Reversal
• Translation of Lueders' "Uber die Zustandsanderung durch den Messprozess"
• Quantum Dynamics as an analog of Conditional Probability
• Quantum Bayesian Nets
• Quantum State Smoothing
• From time-reversal symmetry to quantum Bayes' rules
• Fluctuation theorems from Bayesian retrodiction
• Axioms for retrodiction: achieving time-reversal symmetry with a prior
• Strong Convexity of Sandwiched Entropies and Related Optimization Problems
• State retrieval beyond Bayes' retrodiction
• Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
• The Near-optimal Performance of Quantum Error Correction Codes
• On the generic increase of observational entropy in isolated systems
• Operational meanings of a generalized conditional expectation in quantum metrology
• Observational entropy with general quantum priors

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• Realizing the Petz Recovery Map on an NMR Quantum Processor
• Exact large deviations and emergent long-range correlations in sequential quantum East circuits
• Tabletop reversibility of phase-covariant operations