ADAPTIVE SUP-NORM ESTIMATION OF THE WIGNER FUNCTION IN NOISY QUANTUM HOMODYNE TOMOGRAPHY

• 成果类型: Article
• 署名作者: Lounici, Karim; Meziani, Katia; Peyre, Gabriel
• 署名单位: University System of Georgia; Georgia Institute of Technology; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Cote d'Azur; Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Institut Polytechnique de Paris; ENSAE Paris; Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); Centre National de la Recherche Scientifique (CNRS); Universite Cote d'Azur; Universite PSL
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1586
• 发表日期: 2018
• 页码: 1318-1351
• 关键词: nonparametric deconvolution; density deconvolution; wavelet deconvolution; laguerre-polynomials; Inverse problems; Optimal Rates; CONVERGENCE; Minimax; matrix

摘要

In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on 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 <= 1, 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 L-infinity-risk over a class of infinitely differentiable functions. We also compute the lower bound for the 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.