Infinite Dimensional Mean-Field Belavkin Equation: Well-posedness and Derivation

TL;DR

This paper establishes the well-posedness and derivation of an infinite-dimensional mean-field Belavkin equation, providing a rigorous framework for analyzing large quantum systems under continuous measurement.

Contribution

It extends mean-field limit derivations to infinite-dimensional quantum systems, overcoming previous bounded operator restrictions and employing fixed-point methods for well-posedness.

Findings

Proves global well-posedness of the infinite-dimensional Belavkin equation.

Derives rigorous mean-field limit for wave functions in $L^2 (R^d)$.

Provides a foundation for simulating and controlling large quantum systems.

Abstract

We analyze the mean-field limit of a stochastic Schr{\"o}dinger equation arising in quantum optimal control and mean-field games, where N interacting particles undergo continuous indirect measurement. For the open quantum system described by Belavkin's filtering equation, we derive a mean-field approximation under minimal assumptions, extending prior results limited to bounded operators and finitedimensional settings. By establishing global well-posedness via fixed-point methods-avoiding measure-change techniques-we obtain higher regularity solutions. Furthermore, we prove rigorous convergence to the mean-field limit in an infinitedimensional framework. Our work provides the first derivation of such limits for wave functions in $L^2 (R^d)$, with implications for simulating and controlling large quantum systems.