From time-reversal symmetry to quantum Bayes' rules

TL;DR

This paper develops a comprehensive framework for quantum Bayes' rules, linking them to state-over-time concepts and time-reversal symmetry, unifying many existing proposals in quantum inference.

Contribution

It introduces a systematic definition of quantum Bayes' rule based on state-over-time and time-reversal symmetry, unifying previous approaches.

Findings

Most existing quantum Bayes' rules are special cases of the new framework.

The framework clarifies the relationship between quantum states over time and symmetry.

Provides a foundation for future quantum inference methods.

Abstract

Bayes' rule $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical inference. Many attempts have been made to extend this rule to quantum systems, the significance of which we are only beginning to understand. In this work, we develop a systematic framework for defining Bayes' rule in the quantum setting, and we show that a vast majority of the proposed quantum Bayes' rules appearing in the literature are all instances of our definition. Moreover, our Bayes' rule is based upon a simple relationship between the notions of state over time and a time-reversal symmetry map, both of which are introduced here.