Axioms for retrodiction: achieving time-reversal symmetry with a prior

Abstract

We propose a category-theoretic definition of retrodiction and use it to exhibit a time-reversal symmetry for all quantum channels. We do this by introducing retrodiction families and functors, which capture many intuitive properties that retrodiction should satisfy and are general enough to encompass both classical and quantum theories alike. Classical Bayesian inversion and all rotated and averaged Petz recovery maps define retrodiction families in our sense. However, averaged rotated Petz recovery maps, including the universal recovery map of Junge-Renner-Sutter-Wilde-Winter, do not define retrodiction functors, since they fail to satisfy some compositionality properties. Among all the examples we found of retrodiction families, the original Petz recovery map is the only one that defines a retrodiction functor. In addition, retrodiction functors exhibit an inferential time-reversal symmetry consistent with the standard formulation of quantum theory. The existence of such a retrodiction functor seems to be in stark contrast to the many no-go results on time-reversal symmetry for quantum channels. One of the main reasons is because such works defined time-reversal symmetry on the category of quantum channels alone, whereas we define it on the category of quantum channels and quantum states. This fact further illustrates the importance of a prior in time-reversal symmetry.

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• Experimental demonstration of input-output indefiniteness in a single quantum device
• Quantum Bayes' rule and Petz transpose map from the minimum change principle
• Bayesian inversion and the Tomita-Takesaki modular group
• Operational meanings of a generalized conditional expectation in quantum metrology
• Optimality Condition for the Petz Map
• The spatiotemporal doubled density operator: a unified framework for analyzing spatial and temporal quantum processes
• Role of Dilations in Reversing Physical Processes: Tabletop Reversibility and Generalized Thermal Operations
• Observational entropy with general quantum priors
• Petz recovery maps for qudit quantum channels
• Quantum reversal: a general theory of coherent quantum absorbers
• Proper and improper mixed states serve as different prior beliefs for quantum state retrodiction
• A Retrodictive Approach to Quantum State Smoothing
• Bipartite quantum states admitting a causal explanation
• Quantifying Irreversibility via Bayesian Subjectivity for Classical & Quantum Linear Maps
• Macroscopicity and observational deficit in states, operations, and correlations