The Annals of Statistics

Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography

Karim Lounici, Katia Meziani, and Gabriel Peyré

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Abstract

In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on $\mathbb{R}^{2}$, which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2<\eta\leq1$, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the $\mathbb{L}_{\infty}$-risk over a class of infinitely differentiable functions. We also compute the lower bound for the $\mathbb{L}_{2}$-risk. We construct an adaptive estimator, that is, which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness of the class of functions up to a logarithmic factor in the sample size. Finite sample behaviour of our adaptive procedure is explored through numerical experiments.

Article information

Source
Ann. Statist., Volume 46, Number 3 (2018), 1318-1351.

Dates
Received: September 2015
Revised: May 2017
First available in Project Euclid: 3 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1525313084

Digital Object Identifier
doi:10.1214/17-AOS1586

Subjects
Primary: 62G05: Estimation 81V80: Quantum optics

Keywords
Nonparametric minimax estimation adaptive estimation inverse problem $\mathbb{L}_{2}$ and $\mathbb{L}_{\infty}$ risks quantum homodyne tomography Wigner function Radon transform quantum state

Citation

Lounici, Karim; Meziani, Katia; Peyré, Gabriel. Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography. Ann. Statist. 46 (2018), no. 3, 1318--1351. doi:10.1214/17-AOS1586. https://projecteuclid.org/euclid.aos/1525313084


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Supplemental materials

  • Supplement to “Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography”. This supplementary material contains proofs of several technical results.