A Quantum Bayes' Rule and Related Inference

TL;DR

This paper develops a quantum analogue of Bayesian inference using the concept of instruments, extending classical Bayesian methods to quantum systems and analyzing their behavior with increasing observations.

Contribution

It introduces a quantum version of Bayes' rule based on instruments, generalizes Bayesian inference to quantum states, and explores the asymptotic behavior of quantum posteriors.

Findings

Quantum Bayes' rule elaborates state updates under observations.

The theory generalizes classical Bayesian inference.

As the number of observations increases, the quantum posterior converges.

Abstract

In this work a quantum analogue of Bayesian inference is considered. Based on the notion of instrument, we propose a quantum analogue of Bayes' rule, which elaborates how a prior normal state updates under observations. Besides, we investigate the limit of posterior normal state as the number of observations goes to infinity. After that, we generalize the fundamental notions and results of Bayesian inference according to quantum Bayes' rule. It is noted that our theory not only retains the classical one as a special case but possesses many new features as well.