An invitation to quantum filtering and smoothing theory based on two inner products (Building Foundations for Quantum Statistical Modeling)

Abstract

The purpose of this paper is to introduce the quantum filtering and a smoothing theory for Markovian open quantum dynamical systems briefly. The filtering and smoothing theory for classical systems are well developed and show their performance used in practice, and the quantum filtering theory has also developed from 1980s based on the quantum version of the classical conditional expectation. However, the quantum smoothing theory has not developed since the quantum conditional expectation is not defined as well. We developed a quantum smoothing theory based on two quantum inner products and show it with the filtering theory. The weak value is naturally defined as the minimum mean square estimate in our frame work and the quantum smoother, the dynamical estimator of weak value, is derived.