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Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data

Peyré, Gabriel; Meziani, Katia; Alquier, Pierre (2013), Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data, Inverse Problems, 29, 7. http://dx.doi.org/10.1088/0266-5611/29/7/075017

Type
Article accepté pour publication ou publié
External document link
http://fr.arxiv.org/abs/1301.7644
Date
2013
Journal name
Inverse Problems
Volume
29
Number
7
Publisher
IOP
Publication identifier
http://dx.doi.org/10.1088/0266-5611/29/7/075017
Metadata
Show full item record
Author(s)
Peyré, Gabriel
Meziani, Katia
Alquier, Pierre
Abstract (EN)
In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2 < η ≤ 1, we propose a novel estimator of a quantum state whose density matrix elements ρm, n decrease like $C{\rm e}^{-B(m+n)^{ r/ 2}}$, for fixed C ≥ 1, B > 0 and 0 < r ≤ 2. In contrast to previous works, we focus on the case where r, C and B are unknown. The procedure estimates the matrix coefficients by a projection method on the pattern functions, and then by soft-thresholding the estimated coefficients. We prove that under the $\mathbb {L}_2$-loss our procedure is adaptive rate-optimal, in the sense that it achieves the same rate of convergence as the best possible procedure relying on the knowledge of (r, B, C). Finite sample behaviour of our adaptive procedure is explored through numerical experiments.
Subjects / Keywords
Wigner function; Radon transform; Thresholded estimato; Quantum homodyne tomography; Projection estimator,; Pattern functions; Non-parametric estimation; Invers problem; L 2 Risk; aussian noise; Density matrix,; Adaptive estimation,

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