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<p>We analyze the mean-field limit of a stochastic Schrödinger equation arising in quantum optimal control and mean-field games, where <i>N</i> 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 finite-dimensional 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 infinite-dimensional framework. Our work provides the first derivation of such limits for wave functions in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2(\mathbb {R}^d)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, with implications for simulating and controlling large quantum systems.</p>

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

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Anne de Bouard,
  • Gaoyue Guo,
  • Théo Hérouard

摘要

We analyze the mean-field limit of a stochastic Schrödinger 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 finite-dimensional 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 infinite-dimensional framework. Our work provides the first derivation of such limits for wave functions in \(L^2(\mathbb {R}^d)\) L 2 ( R d ) , with implications for simulating and controlling large quantum systems.