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June 2018 Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography
Karim Lounici, Katia Meziani, Gabriel Peyré
Ann. Statist. 46(3): 1318-1351 (June 2018). DOI: 10.1214/17-AOS1586

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.

Citation

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Karim Lounici. Katia Meziani. Gabriel Peyré. "Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography." Ann. Statist. 46 (3) 1318 - 1351, June 2018. https://doi.org/10.1214/17-AOS1586

Information

Received: 1 September 2015; Revised: 1 May 2017; Published: June 2018
First available in Project Euclid: 3 May 2018

zbMATH: 1393.62136
MathSciNet: MR3798005
Digital Object Identifier: 10.1214/17-AOS1586

Subjects:
Primary: 62G05 , 81V80

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

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 3 • June 2018
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