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

V. N. Kolokoltsov

Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 1, pp. 97-121

Résumé

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 in a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Schrödinger equations or Hartree equations, or Gross–Pitaevskii equations. In this paper, we extend some of these convergence results to a stochastic framework. Specifically, 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 regarded 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.

Détail

Keywords: quantum dynamic law of large numbers, quantum filtering, homodyne detection, nonlinear stochastic Schrödinger equation, quantum interacting particles, quantum control, quantum mean-field games, infinite-dimensional McKean–Vlasov diffusion on manifold.
Mots-clés : Belavkin equation

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