On quantum Stochastic Master equations

TL;DR

This paper develops a rigorous mathematical framework for quantum stochastic master equations in infinite-dimensional spaces and extends the theory to quantum mean-field games involving McKean-Vlasov diffusions.

Contribution

It provides the first rigorous solution for infinite-dimensional quantum stochastic master equations with bounded operators and establishes well-posedness for quantum McKean-Vlasov diffusions.

Findings

Rigorous solution for infinite-dimensional quantum stochastic master equations.

Well-posedness results for quantum McKean-Vlasov diffusions.

Extension of quantum filtering theory to new classes of stochastic equations.

Abstract

Stochastic Master equations or quantum filtering equations for mixed states are well known objects in quantum physics. Building a mathematically rigorous theory of these equations in infinite-dimensional spaces is a long standing open problem. The first objective of this paper is to give a solution to this problem under the assumption of bounded operators providing coupling with environment (or a measurement devise). Furthermore, recently the author built the theory of the law of large number limit for continuously observed interacting quantum particle systems leading to quantum mean-field games. These limits are described by certain nontrivial extensions of quantum stochastic master equations that can be looked at as infinite-dimensional operator-valued McKean-Vlasov diffusions. The second objective of this paper is to provide a well-posedness result for these new class of McKean-Vlasov diffusions.