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

TL;DR

This paper introduces a category-theoretic framework for retrodiction in quantum and classical theories, establishing a time-reversal symmetry via retrodiction functors, with the Petz recovery map as a key example.

Contribution

It defines retrodiction families and functors, demonstrating that only the Petz recovery map forms a retrodiction functor, and highlights the role of priors in quantum time-reversal symmetry.

Findings

Petz recovery map uniquely defines a retrodiction functor.

Retrodiction functors exhibit a consistent time-reversal symmetry.

A prior is essential for defining time-reversal symmetry in quantum channels.

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.