Optimal Nonlinear Filtering of Quantum State

The aim of this paper is to give an interpretation of the linear or nonlinear filtration problem as a quantization problem. Based on this, we extend the optimal filtering equation known from the Stratonovich filtering theory on the quantum process case. In contrast to the Kalman filter often used in such problems, the Stratonovich equation provides the optimal solution for nonlinear observation models. The observation model is based on an indirect measurement method, where the measurement is performed on an ancilla system that is interacted with an unknown one. The latter model is interpreted in case of a single-qudit system. Since the measured random variables and experiment are fundamentally different from each other, the obtained single-qudit observation model has a substantially different physical nature from the two-qubit one. However, in both problems, the Stratonovich optimal filter equation is equally applicable.