Skip to content
DraconiticResearch
Search Sources API
178,970 studies31 sourcesUpdated 29 min ago
Peer-reviewedCC BY 4.0Axioms

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

  • Vladimir Orlov1
  • Liubov Markovich2,3
  1. 1Russian Quantum Center, Skolkovo, Moscow 121205, Russia
  2. 2Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
  3. 3〈aQaL〉 Applied Quantum Algorithms, 2333 CA Leiden, The Netherlands
Open at sourceDOICite

Abstract

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 that KQSE is competitive with 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.

Related

  1. Efficient estimation methods for non-Gaussian regression models in continuous time
    Evgeny A Pchelintsev, Sergey Pergamenshchikov, Maria Povzun
    Annals of the Institute of Statistical MathematicsPeer-reviewedPublisher termsFull text5 citedStatisticsHAL
  2. Bayesian Estimation of Continuous Brain Connectivity with Log-Gaussian Cox Processes
    Jaehoan Kim, William Consagra, Debdeep Pati +1
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  3. Nonparametric hazard rate estimation with associated kernels and minimax bandwidth choice
    Luce Breuil, Sarah Kaakai
    HALPreprintLicence Ouverte 2.0Full textStatisticsHAL
  4. GNet: A scalable and flexible Gaussian process network with nonparametric neurons
    Mengyang Gu
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  5. nethist: An R package for Nonparametric Graphon Estimation via Network Histograms
    Youngseok Song, Sofia C. Olhede
    arXivPreprintCC BY 4.0Full textStatisticsarXiv

Cite

BibTeXRIS
BibTeX
@article{orlov2026optimal,
  title = {{Optimal Nonparametric Estimation of Phase-Space Representations for Non-Gaussian Continuous Variable Quantum States}},
  author = {Vladimir Orlov and Liubov Markovich},
  journal = {Axioms},
  year = {2026},
  month = nov,
  doi = {10.3390/axioms15110815},
  url = {https://www.mdpi.com/2075-1680/15/11/815},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Optimal Nonparametric Estimation of Phase-Space Representations for Non-Gaussian Continuous Variable Quantum States
AU  - Vladimir Orlov
AU  - Liubov Markovich
PY  - 2026
DA  - 2026/11/02
JO  - Axioms
DO  - 10.3390/axioms15110815
UR  - https://www.mdpi.com/2075-1680/15/11/815
AB  - 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 that KQSE is competitive with 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.
SN  - 2075-1680
ER  -