A Quantum Bayes' Rule and Related Inference

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.