On the Mathematical Theory of Quantum Stochastic Filtering Equations for Mixed States
Vassili N Kolokoltsov
TL;DR
The paper develops a rigorous mathematical framework for quantum stochastic filtering equations governing mixed states in infinite-dimensional settings with unbounded coupling operators. It extends prior bounded-L results by adopting Mora-Rebolledo-type hypotheses, establishing well-posedness for both linear and nonlinear Belavkin equations in pure and mixed states, and formulating the linear theory within the Hilbert-Schmidt space before translating to normalized nonlinear dynamics. Key contributions include strong and weak solvability results for pure-state dynamics, a detailed analysis of linear stochastic master equations, and a robust normalization-based treatment of mixed-state filtering, supported by finite-dimensional approximation methods. The work lays a solid foundation for deriving quantum filtering equations from measurement limits and for connecting stochastic quantum filtering to quantum mean-field limits and feedback control in infinite dimensions.
Abstract
Quantum filtering equations for mixed states were developed in 80th of the last century. Since then the problem of building a rigorous mathematical theory for these equations in the basic infinite-dimensional settings has been a challenging open mathematical problem. In a previous paper, the author developed the theory of these equations in the case of bounded coupling operators, including a new version that arises as the law of large numbers for interacting particles under continuous observation and thus leading to the theory of quantum mean field games. In this paper, the main body of these results is extended to the basic cases of unbounded coupling operators.