Nous nous intéressons à un problème de statistique non-paramétrique issu de la physique, et plus précisément à la tomographie quantique, c'est-à-dire la détermination de l'état quantique d'un mode de la lumière via une mesure homodyne. Nous appliquons plusieurs procédures de sélection de modèles : des estimateurs par projection pénalisés, où on peut utiliser soit des fonctions motif, soit des ondelettes, et l'estimateur du maximum de vraisemblance pénalisé. Dans chaque cas, nous obtenons une inégalité oracle. Nous prouvons également une vitesse de convergence polynomiale pour ce problème non-paramétrique, pour les estimateurs par projection. Nous appliquons ensuite des idées à la calibration d'un photocompteur, l'appareil dénombrant le nombre de photons dans un rayon lumineux. Le problème mathématique se réduit dans ce cas à un problème non-paramétrique à données manquantes. Nous obtenons à nouveau des inégalités oracle, qui nous assurent des vitesses de convergence d'autant meilleures que le photocompteur est bon.
This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators, where we may use pattern functions or wavelets, and penalized maximum likelihood estimators. In all these cases, we get oracle inequalities. In the former we also have a polynomial rate of convergence for the non-parametric problem. We finish the paper with applications of similar ideas to the calibration of a photocounter, a measurement apparatus counting the number of photons in a beam. Here the mathematical problem reduces similarly to a non-parametric missing data problem. We again get oracle inequalities, and better speed if the photocounter is good.
Mots clés : density matrix, model selection, pattern functions estimator, penalized maximum likelihood estimator, penalized projection estimators, quantum calibration, quantum tomography, wavelet estimator, Wigner function
@article{PS_2009__13__363_0,
author = {Kahn, Jonas},
title = {Model selection for quantum homodyne tomography},
journal = {ESAIM: Probability and Statistics},
pages = {363--399},
publisher = {EDP-Sciences},
volume = {13},
year = {2009},
doi = {10.1051/ps:2008017},
mrnumber = {2554961},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps:2008017/}
}
Kahn, Jonas. Model selection for quantum homodyne tomography. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 363-399. doi : 10.1051/ps:2008017. http://archive.numdam.org/articles/10.1051/ps:2008017/
[1] , and , An invitation to quantum tomography. J. Royal Statist. Soc. B 67 (2005) 109-134. | MR | Zbl
[2] , , and , Maximum-likelihood estimation of the density matrix. Phys. Rev. A 61 (1999) R010304.
[3] , and , On quantum statistical inference (with discussion). J. R. Statist. Soc. B 65 (2003) 775-816. | MR | Zbl
[4] , On a modification of Chebyshev's inequality and of the error formula of Laplace. In Collected works, Vol. 4 (1964).
[5] , and , Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data. Ann. Statist. 35 (2007) 465-494. | MR | Zbl
[6] and , Poisson intensity estimation for tomographic data using a wavelet shrinkage approach. IEEE Trans. Inf. Theory 48 (2002) 2794-2802. | MR | Zbl
[7] , and , Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 50 (1994) 4298-4302.
[8] , and , Homodyne detection of the density matrix of the radiation field. Phys. Rev. A 52 (1995) R1801-R1804.
[9] , , , Quantum calibration of measuring apparatuses. Phys. Rev. Lett. 93 (2004) 250407.
[10] , The Radon transform and some of its applications. John Wiley & Sons, New York (1983). | MR | Zbl
[11] , Higher Transcendental Functions, Vol. 2. McGraw-Hill (1953). | Zbl
[12] , Quantum Asymptotics, volume 36 of IMS Lect. Notes-Monograph Ser. (2001) 255-285. | MR
[13] , Quantum Detection and Estimation Theory. Academic Press, New York (1976).
[14] , Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58 (1964) 13-30. | MR | Zbl
[15] , Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Publishing Company, Amsterdam (1982). | MR | Zbl
[16] , Sélection de modèles en tomographie quantique. Master's thesis, École Normale Supérieure, Université Paris-Sud, 2004.
[17] , Measuring the Quantum State of Light. Cambridge University Press (1997).
[18] , and , Tomographic reconstruction of the density matrix via pattern functions. Phys. Rev. A 52 (1995) 4899-4907.
[19] , Concentration Inequalities and Model Selection. École d'été de Probabilité de Saint-Flour, 2003. Lect. Notes Math. Springer-Verlag, Berlin (2006). | MR | Zbl
[20] , , and , Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. Phys. Rev. Lett. 70 (1993) 1244-1247.
[21] and , Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40 (1989) 2847-2849.
Cité par Sources :
