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arXiv:2506.12249

Graphon Quantum Filtering Systems

Hamed Amini, Nina H. Amini, Sofiane Chalal, Gaoyue Guo

Published Jun 13, 2025• 0 citations
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Abstract

We consider a non-exchangeable system of interacting quantum particles with mean-field type interactions, subject to continuous measurement on dense graphs. In the mean-field limit, we derive a graphon-based quantum filtering system, establish its well-posedness, and prove propagation of chaos for multi-class bosonic systems with blockwise interactions. We then discuss applications to quantum state preparation and quantum graphon games.

DOI
10.48550/arxiv.2506.12249

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We consider a non-exchangeable system of interacting quantum particles with mean-field type interactions, subject to continuous measurement on dense graphs. In the mean-field limit, we derive a graphon-based quantum filtering system, establish its well-posedness, and prove propagation of chaos for multi-class bosonic systems with blockwise interactions. We then discuss applications to quantum state preparation and quantum graphon games.
The theory of quantum filtering (of quantum continuous measurements) was developed by V.P. Belavkin about 40 years ago. Since then it attracted attention of numerous investigators including mathematicians, theoretical and experimental physicists. However, the rigorous mathematical theory of the filtering equations for mixed states in basic infinite-dimensional quantum systems remained an open problem, which was resolved by the author recently. This survey paper presents in full the mathematical theory of quantum filtering equations, their rigorous derivation from basic principles, the corresponding law of large number limits (propagation of chaos) and related topics. Applications to feedback control, quantum dynamic and mean-field games are discussed.
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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 has been 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 device). 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.
Quantum games and mean-field games (MFG) represent two important new branches of game theory. In a recent paper the author developed quantum MFGs merging these two branches. These quantum MFGs were based on the theory of continuous quantum observations and filtering of diffusive type. In the present paper we develop the analogous quantum MFG theory based on continuous quantum observations and filtering of counting type. However, proving existence and uniqueness of the solutions for resulting limiting forward-backward system based on jump-type processes on manifolds seems to be more complicated than for diffusions. In this paper we only prove that if a solution exists, then it gives an ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.
We use probabilistic and stochastic-analysis methods to prove trace-norm propagation of chaos for finite-dimensional quantum mean-field systems governed by Belavkin equations. The particles are density matrices, interact through a mean-field Hamiltonian, and are continuously monitored through independent diffusive observation channels. The limiting dynamics is a nonlinear matrix-valued McKean-Vlasov diffusion, random through its local observation record and coupled through the deterministic averaged state. The main result treats arbitrary one-particle density matrices, including mixed states, and both perfect and inefficient measurement regimes. Under strong tensorization of the initial data, every fixed marginal converges uniformly on compact time intervals to the tensor product of the nonlinear limiting filters, with an explicit quantitative bound. The proof combines purification, fully observed dilation, conditional expectation, relative entropy, and uniform stability of the associated Zakai equations. In the skew-adjoint measurement case, exterior observation noises disappear from the marginal equations and a stochastic BBGKY hierarchy is recovered. Under only marginal chaoticity of permutation-invariant initial states, we prove convergence of fixed marginals by an iteration of this hierarchy.