Proper and improper mixed states serve as different prior beliefs for quantum state retrodiction
arXiv:2502.10030 · doi:10.1103/xx43-p1py
Abstract
A mixed quantum state can be taken as capturing an unspecified form of ignorance; or as describing the lack of knowledge about the true pure state of the system ("proper mixture"); or as arising from entanglement with another system that has been disregarded ("improper mixture"). These different views yield identical density matrices and therefore identical predictions for future measurements. But when used as prior beliefs for inferring the past state from later observations ("retrodiction"), they lead to different updated beliefs. This is a purely quantum feature of Bayesian agency. Based on this observation, we establish a framework for retrodicting on any quantum belief and we prove a necessary and sufficient condition for the equivalence of beliefs. We also illustrate how these differences have operational consequences in quantum state recovery.
updated to the published version
References in corpus (29)
- Informational derivation of Quantum Theory
- Probabilistic theories with purification
- Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
- Reversing quantum dynamics with near-optimal quantum and classical fidelity
- Quantum Bayes rule
- Quantum Graphical Models and Belief Propagation
- Past quantum states
- Recoverability in quantum information theory
- Universal recovery maps and approximate sufficiency of quantum relative entropy
- Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map
- Bayes' theorem and quantum retrodiction
- Quantum algorithm for Petz recovery channels and pretty good measurements
- Quantum Bayesian Nets
- Quantum State Smoothing
- Fluctuation Theorems for a Quantum Channel
- Universal recovery map for approximate Markov chains
- Master Equation for Retrodiction of Quantum Communication Signals
- From time-reversal symmetry to quantum Bayes' rules
- Fluctuation theorems from Bayesian retrodiction
- Reversing Lindblad Dynamics via Continuous Petz Recovery Map
- Axioms for retrodiction: achieving time-reversal symmetry with a prior
- Retrodiction of Generalised Measurement Outcomes
- State retrieval beyond Bayes' retrodiction
- The Near-optimal Performance of Quantum Error Correction Codes
- Reversibility conditions for quantum channels and their applications
- Noise-adapted recovery circuits for quantum error correction
- Quantum Bayes' rule and Petz transpose map from the minimum change principle
- Retrodiction of measurement outcomes on a single quantum system reveals entanglement with its environment
- A Retrodictive Approach to Quantum State Smoothing