The quantum filtering problem as a dynamical covariance condition

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 $Γ^{(1)}_t$ is compared with the one-particle filter $γ_t$ driven by its own innovation, the mean-field Hamiltonian evaluated at the deterministic Hartree-Lindblad mean $η_t$ . The fluctuation $F^N_t=\sqrt{N}(Γ^{(1)}_t-γ_t)$ 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 $\hat G^N_t=\sqrt{N}(Γ_t-η_t)$ , 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.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 $Γ^{(1)}_t$ is compared with the one-particle filter $γ_t$ driven by its own innovation, the mean-field Hamiltonian evaluated at the deterministic Hartree-Lindblad mean $η_t$. The fluctuation $F^N_t=\sqrt{N}(Γ^{(1)}_t-γ_t)$ 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 $\hat G^N_t=\sqrt{N}(Γ_t-η_t)$, 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.

arXiv:2609.13346Sep 11, 20260 citations

Non-Commuting Covariance Geometry in Filter-Integrated Dynamical-Decoupling Spectroscopy

Hiroyuki Shioiri

This paper studies the covariance geometry of filter-integrated observables in dynamical-decoupling spectroscopy. We show that the covariance admits a non-commuting additive decomposition,Ctotal=λCflat+CdevC^{\rm total}=\lambda C^{\rm flat}+C^{\rm dev}Ctotal=λCflat+Cdev,where CflatC^{\rm flat}Cflat represents flat filter-overlap geometry and CdevC^{\rm dev}Cdev encodes spectral deformation induced by power-law noise weights. Numerical analysis of CPMG filters reveals persistent but diluted non-commutativity under protocol enlargement, selective intermediate-mode hybridization, and horizon-precursor growth of covariance ill-conditioning.This paper studies the covariance geometry of filter-integrated observables in dynamical-decoupling spectroscopy. We show that the covariance admits a non-commuting additive decomposition,Ctotal=λCflat+CdevC^{\rm total}=\lambda C^{\rm flat}+C^{\rm dev}Ctotal=λCflat+Cdev,where CflatC^{\rm flat}Cflat represents flat filter-overlap geometry and CdevC^{\rm dev}Cdev encodes spectral deformation induced by power-law noise weights. Numerical analysis of CPMG filters reveals persistent but diluted non-commutativity under protocol enlargement, selective intermediate-mode hybridization, and horizon-precursor growth of covariance ill-conditioning.

May 24, 20260 citations

Quantum Filtering for Linear Dynamical Quantum Systems

Hendra I. Nurdin, Naoki Yamamoto

Communications and control engineering/Communications and control engineering seriesJan 1, 20171 citation

Quantum Filtering and Optimal Control

V. P. Belavkin, Simon Edwards

Quantum Stochastics and InformationAug 1, 200840 citations

Coherent quantum filtering for physically realizable linear quantum plants

Igor G. Vladimirov, Ian R. Petersen

The paper is concerned with a problem of coherent (measurement-free) filtering for physically realizable (PR) linear quantum plants. The state variables of such systems satisfy canonical commutation relations and are governed by linear quantum stochastic differential equations, dynamically equivalent to those of an open quantum harmonic oscillator. The problem is to design another PR quantum system, connected unilaterally to the output of the plant and playing the role of a quantum filter, so as to minimize a mean square discrepancy between the dynamic variables of the plant and the output of the filter. This coherent quantum filtering (CQF) formulation is a simplified feedback-free version of the coherent quantum LQG control problem which remains open despite recent studies. The CQF problem is transformed into a constrained covariance control problem which is treated by using the Frechet differentiation of an appropriate Lagrange function with respect to the matrices of the filter.The paper is concerned with a problem of coherent (measurement-free) filtering for physically realizable (PR) linear quantum plants. The state variables of such systems satisfy canonical commutation relations and are governed by linear quantum stochastic differential equations, dynamically equivalent to those of an open quantum harmonic oscillator. The problem is to design another PR quantum system, connected unilaterally to the output of the plant and playing the role of a quantum filter, so as to minimize a mean square discrepancy between the dynamic variables of the plant and the output of the filter. This coherent quantum filtering (CQF) formulation is a simplified feedback-free version of the coherent quantum LQG control problem which remains open despite recent studies. The CQF problem is transformed into a constrained covariance control problem which is treated by using the Frechet differentiation of an appropriate Lagrange function with respect to the matrices of the filter.

arXiv:1301.3154European Control Conference (ECC)Jul 1, 201327 citations

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

Γ^{(1)}_t

is compared with the one-particle filter

γ_t

driven by its own innovation, the mean-field Hamiltonian evaluated at the deterministic Hartree-Lindblad mean

η_t

. The fluctuation

F^N_t=\sqrt{N}(Γ^{(1)}_t-γ_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}(Γ_t-η_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.