A quantum Bayes’ rule and related inference

A quantum analogue of Bayesian inference is considered here. Quantum state-update rule associated with instrument is elected as a quantum Bayes’ rule. A sufficient condition on the spectrum of a trivial CP instrument repeatedly applied for the uniformly exponential convergence of posterior normal state and a sufficient condition on the kernel of trivial CP instruments sampled from an ensemble as well as the strict positivity of the sequential measurement scheme for the exponential convergence of posterior normal state are obtained, as a result of which, two sufficient conditions for the weak consistency of posterior normal state are deduced as corollaries. The fundamental notions and results of Bayesian inference such as Bayes solution, posterior solution, admissibilty and minimax of Bayes solution, and posterior inference, are generalized based on quantum Bayes’ rule. Our theory retains the classical one as a special case though we note that unlike the classical case, posterior normal state varies with the order of observations; posterior normal state may not converge as the number of observations tends to infinity; for a given quantum Bayesian decision problem, a quantum Bayes solution and a quantum posterior solution are generally no longer equivalent.