A Dynamic Central Limit Theorem for Mean-Field Quantum Filtering

RESUMO

We study Open Problem 3 of Kolokoltsov's survey "Quantum filtering and propagation of chaos for open quantum systems" (2026) on $H=\mathbb{C}^d$ ($2\le d<\infty$ fixed): a dynamic central limit theorem for $N$ mean-field coupled quantum particles under continuous diffusive (homodyne) measurement. The tagged conditional state $\Gamma^{(1)}_t$ is compared with the one-particle filter $\gamma_t$ driven by its own innovation, the mean-field Hamiltonian evaluated at the deterministic Hartree-Lindblad mean $\eta_t$. The fluctuation $F^N_t=\sqrt{N}(\Gamma^{(1)}_t-\gamma_t)$ is not autonomous in the limit: it is driven through the linearised filter by its own innovation, by an idiosyncratic Gaussian martingale carried by the connected correlations with the other particles, and by the empirical field $\hat G^N_t=\sqrt{N}(\Gamma_t-\eta_t)$, which itself satisfies an autonomous linear Gaussian equation, so that $(F,\hat G)$ solves a closed system. Unconditionally, $(F^N,\hat G^N)$ is tight, every limit point solves this system driven by a Brownian motion orthogonal to a square-integrable martingale pair, and the mean covariance of the idiosyncratic martingale is identified as the limit of the rescaled pair correlations. The remaining clauses, concerning signs rather than amplitudes, reduce to a single decorrelation statement fixing the law of the driving pair. It survives in a conditional form, with the common response to the empirical field and the tagged-slot component projected out; without those projections it is false for generic data, which is what fixes them. The conditional form is proved at leading order on a shrinking horizon and from below at its exact orders on every horizon; it is needed only for the concentration of the driving covariances, the Gaussianity of the pair, and uniqueness, and what remains open is one-sided and above the leading order.