Abstract Quantum state estimation is a central task in quantum information theory, aiming to reconstruct an unknown density matrix from a finite number of measurement outcomes. Conventional approaches such as kernel smoothing or maximum likelihood estimation (MLE) often rely on restrictive parametric assumptions and become computationally prohibitive for high-dimensional quantum systems. In this work, we introduce a nonparametric quantum state estimation framework based on the k-Nearest Neighbor (k-NN) approach. The method leverages the local geometric structure of measurement data within the quantum feature space to infer the underlying quantum state without any predefined model. Using intrinsic quantum distance metrics—such as the trace distance and quantum fidelity—the proposed estimator identifies neighboring states and performs a weighted aggregation, followed by a positive-semidefinite projection to ensure physical admissibility. We establish the consistency and asymptotic convergence of the NN-based estimator under mild regularity conditions. Numerical simulations on single- and two-qubit systems confirm that the proposed method achieves high reconstruction accuracy, robustness to measurement noise, and superior computational efficiency compared to kernel- and likelihood-based estimators. Overall, this study highlights the potential of data-driven, geometry-aware, nonparametric learning for scalable quantum state reconstruction, providing a promising alternative for modern quantum information processing.Abstract Quantum state estimation is a central task in quantum information theory, aiming to reconstruct an unknown density matrix from a finite number of measurement outcomes. Conventional approaches such as kernel smoothing or maximum likelihood estimation (MLE) often rely on restrictive parametric assumptions and become computationally prohibitive for high-dimensional quantum systems. In this work, we introduce a nonparametric quantum state estimation framework based on the k-Nearest Neighbor (k-NN) approach. The method leverages the local geometric structure of measurement data within the quantum feature space to infer the underlying quantum state without any predefined model. Using intrinsic quantum distance metrics—such as the trace distance and quantum fidelity—the proposed estimator identifies neighboring states and performs a weighted aggregation, followed by a positive-semidefinite projection to ensure physical admissibility. We establish the consistency and asymptotic convergence of the NN-based estimator under mild regularity conditions. Numerical simulations on single- and two-qubit systems confirm that the proposed method achieves high reconstruction accuracy, robustness to measurement noise, and superior computational efficiency compared to kernel- and likelihood-based estimators. Overall, this study highlights the potential of data-driven, geometry-aware, nonparametric learning for scalable quantum state reconstruction, providing a promising alternative for modern quantum information processing.

Mar 30, 20260 citations

Optimal Nonparametric Estimation of Phase-Space Representations for Non-Gaussian Continuous Variable Quantum States

Vladimir A. Orlov, Liubov A. Markovich

We further develop Kernel Quantum State Estimation (KQSE), a fully data-driven nonparametric method for continuous variable quantum state reconstruction and characterization, introduced in our recent work and rooted in nonparametric kernel density estimation (KDE). Unlike approaches relying on finite-dimensional basis truncations, parametric ansätze, or prior models, KQSE combines a new kernel estimator of the characteristic function of the tomographic quadrature distribution with suitable kernel integral transformations. The characteristic function is estimated directly from experimental homodyne or heterodyne data and subsequently transformed to estimate quantum state representations and characteristics, including the Wigner function and the density matrix kernel. Observing that several other physically relevant representations and characteristics admit transformations of a closely related form, we derive convergence rates for the corresponding broad class of KQSE-based estimators. The resulting framework covers all phase-space representations, the photon-number tomogram, trace products of quantum states, and purity. We derive mean squared error convergence rates for these kernel transformed estimators and show that the corresponding KQSE applications inherit the near optimal rate O(T^{-1}), where $T$ is the total number of measurements. The proposed theory applies equally to Gaussian and non-Gaussian continuous variable quantum states without imposing a Fock space cutoff. Numerical experiments on simulated Gaussian and non-Gaussian states and real homodyne data demonstrate the advantages of KQSE over state-of-the-art methods, establishing it as a statistically consistent and computationally modest framework for continuous variable quantum state estimation and characterization, particularly in the non-Gaussian regime.We further develop Kernel Quantum State Estimation (KQSE), a fully data-driven nonparametric method for continuous variable quantum state reconstruction and characterization, introduced in our recent work and rooted in nonparametric kernel density estimation (KDE). Unlike approaches relying on finite-dimensional basis truncations, parametric ansätze, or prior models, KQSE combines a new kernel estimator of the characteristic function of the tomographic quadrature distribution with suitable kernel integral transformations. The characteristic function is estimated directly from experimental homodyne or heterodyne data and subsequently transformed to estimate quantum state representations and characteristics, including the Wigner function and the density matrix kernel. Observing that several other physically relevant representations and characteristics admit transformations of a closely related form, we derive convergence rates for the corresponding broad class of KQSE-based estimators. The resulting framework covers all phase-space representations, the photon-number tomogram, trace products of quantum states, and purity. We derive mean squared error convergence rates for these kernel transformed estimators and show that the corresponding KQSE applications inherit the near optimal rate O(T^{-1}), where $T$ is the total number of measurements. The proposed theory applies equally to Gaussian and non-Gaussian continuous variable quantum states without imposing a Fock space cutoff. Numerical experiments on simulated Gaussian and non-Gaussian states and real homodyne data demonstrate the advantages of KQSE over state-of-the-art methods, establishing it as a statistically consistent and computationally modest framework for continuous variable quantum state estimation and characterization, particularly in the non-Gaussian regime.

quant-pharXiv:2608.22379Aug 25, 20260 citations

Preliminaries of Quantum-State Estimation

WORLD SCIENTIFIC eBooksAug 12, 20150 citations

Lectures on Quantum State Estimation

Berthold‐Georg Englert

WORLD SCIENTIFIC eBooksMay 14, 20250 citations

Bayesian nonparametric estimation for Quantum Homodyne Tomography

Zacharie Naulet, Éric Barat

We estimate the quantum state of a light beam from results of quantum homodyne tomography noisy measurements performed on identically prepared quantum systems. We propose two Bayesian nonparametric approaches. The first approach is based on mixture models and is illustrated through simulation examples. The second approach is based on random basis expansions. We study the theoretical performance of the second approach by quantifying the rate of contraction of the posterior distribution around the true quantum state in the $L^{2}$ metric.We estimate the quantum state of a light beam from results of quantum homodyne tomography noisy measurements performed on identically prepared quantum systems. We propose two Bayesian nonparametric approaches. The first approach is based on mixture models and is illustrated through simulation examples. The second approach is based on random basis expansions. We study the theoretical performance of the second approach by quantifying the rate of contraction of the posterior distribution around the true quantum state in the $L^{2}$ metric.

arXiv:1610.01895Electronic Journal of StatisticsJan 1, 20172 citations

Quantum state and process estimation

Springer eBooksJan 1, 20070 citations

T

is the total number of measurements. The proposed theory applies equally to Gaussian and non-Gaussian continuous variable quantum states without imposing a Fock space cutoff. Numerical experiments on simulated Gaussian and non-Gaussian states and real homodyne data demonstrate the advantages of KQSE over state-of-the-art methods, establishing it as a statistically consistent and computationally modest framework for continuous variable quantum state estimation and characterization, particularly in the non-Gaussian regime.

L^{2}

metric.