The law of large numbers for quantum stochastic filtering and control of many particle systems

TL;DR

This paper extends the law of large numbers to stochastic quantum systems, deriving a new type of nonlinear diffusion equation for large ensembles of interacting particles under quantum filtering.

Contribution

It introduces a stochastic framework for quantum particle systems, deriving a novel infinite-dimensional nonlinear diffusion equation as a limit of Belavkin filtering.

Findings

Established convergence to a new nonlinear diffusion equation

Extended classical large number laws to stochastic quantum settings

Provided mathematical foundation for quantum mean-field games

Abstract

There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing the limiting individual behavior of particles inside a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Scr\"odinger equations or Hartree equations, or Gross-Pitaevski equations. In this paper we extend some of these convergence results to a stochastic framework. Concretely we work with the Belavkin stochastic filtering of many particle quantum systems. The resulting limiting equation is an equation of a new type, which can be seen as a complex-valued infinite dimensional nonlinear diffusion of McKean-Vlasov type. This result is the key ingredient for the theory of quantum mean-field games developed by the author in a previous paper.