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Graphon Quantum Filtering Systems

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

TL;DR

The paper develops a graphon-based mean-field framework for continuously observed, heterogeneous quantum systems, addressing non-exchangeable interactions. It establishes well-posedness of graphon Belavkin filtering equations on a Fubini extension and proves propagation of chaos in block-wise bosonic systems, enabling a tractable limiting description with representative blocks. This limiting graphon system is then applied to quantum state preparation and graphon-based quantum dynamic games, illustrating both stabilization via feedback and game-theoretic formulations. Collectively, the results advance mean-field theory for non-exchangeable open quantum systems and pave the way for scalable control and strategic analysis in large quantum networks.

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.

Graphon Quantum Filtering Systems

TL;DR

The paper develops a graphon-based mean-field framework for continuously observed, heterogeneous quantum systems, addressing non-exchangeable interactions. It establishes well-posedness of graphon Belavkin filtering equations on a Fubini extension and proves propagation of chaos in block-wise bosonic systems, enabling a tractable limiting description with representative blocks. This limiting graphon system is then applied to quantum state preparation and graphon-based quantum dynamic games, illustrating both stabilization via feedback and game-theoretic formulations. Collectively, the results advance mean-field theory for non-exchangeable open quantum systems and pave the way for scalable control and strategic analysis in large quantum networks.

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.

Paper Structure

This paper contains 19 sections, 14 theorems, 190 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum filtering
  4. Graphon theory
  5. Fubini extension theory
  6. Non-exchangeable many-body observed quantum particles
  7. Mean-field graphon quantum filtering systems
  8. Propagation of chaos: Block-wise case
  9. Limiting system
  10. Propagation of chaos
  11. Application
  12. Quantum state preparation
  13. Asymptotic behavior without control
  14. Deterministic state preparation
  15. Simulations
  16. ...and 4 more sections

Key Result

Proposition 2.2

If the measurement efficiency is perfect, i.e., $\eta = 1$, and the initial state is pure $(\rho_0 = \ket{\psi_0}\bra{\psi_0})$, then purity is preserved and the dynamics belavkinhomodyne admit the following pure-state representation: or, equivalently,

Figures (2)

  • Figure 1: Asymptotic behavior under $\underline{\alpha} \equiv 0$.
  • Figure 2: Quantum state preparation using the control law \ref{['eq:feedback-law']}.

Theorems & Definitions (32)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3: Step Kernel
  • Definition 2.4: Essential pairwise independence
  • Definition 2.5: Fubini extension
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Lemma 4.1: kolokoltsov22qmfg
  • ...and 22 more