Nonparametric goodness-of fit testing in quantum homodyne tomography with noisy data

In the framework of quantum optics, we study the problem of goodness-of-fit testing in a severely ill-posed inverse problem. A novel testing procedure is introduced and its rates of convergence are investigated under various smoothness assumptions. The procedure is derived from a projection-type estimator, where the projection is done in $\mathbb{L}_2$ distance on some suitably chosen pattern functions. The proposed methodology is illustrated with simulated data sets.Comment: Published in at http://dx.doi.org/10.1214/08-EJS286 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Rank penalized estimation of a quantum system

We introduce a new method to reconstruct the density matrix $\rho$ of a system of $n$-qubits and estimate its rank $d$ from data obtained by quantum state tomography measurements repeated $m$ times. The procedure consists in minimizing the risk of a linear estimator $\hat{\rho}$ of $\rho$ penalized by given rank (from 1 to $2^n$), where $\hat{\rho}$ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting density matrix associated to this rank. We establish an upper bound for the error of penalized estimator, evaluated with the Frobenius norm, which is of order $dn(4/3)^n /m$ and consistency for the estimator of the rank. The proposed methodology is computationaly efficient and is illustrated with some example states and real experimental data sets

State estimation in quantum homodyne tomography with noisy data

In the framework of noisy quantum homodyne tomography with efficiency parameter $0 < \eta \leq 1$, we propose two estimators of a quantum state whose density matrix elements $\rho_{m,n}$ decrease like $e^{-B(m+n)^{r/ 2}}$, for fixed known $B>0$ and $0<r\leq 2$. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for $0<\eta \leq 1/2$), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in $\mathbb{L}_2$ risk

Estimations et tests non paramétriques en tomographie quantique homodyne

In the setting of quantum optics, the reconstruction of the quantum state (Wigner function or infinite-dimensional density matrix) of a light beam can be seen as a statistical severely ill-posed inverse problem. First, we propose estimators of the density matrix and the Wigner function respectively, using \textit{pattern} functions in the first case and kernel functions in the second. We assume that the unknown density matrix belongs to a nonparametric class which corresponds to typical states prepared in the laboratory. We translate these classes in terms of properties of the associated Wigner function. Second, we estimate the integrated squared Wigner function by a kernel-based second order U-statistic on a larger regularity class. This quadratic functional is a physical measure of the purity of the state. We deduce an adaptive estimator for the Wigner function that does not depend on the smoothness parameters. In the last part of the thesis, we are interested in the problem of goodness-of-fit testing. We give a testing procedure derived from a projection-type estimator on \textit{pattern} functions. We study the upper bounds of the minimax risk for all our procedures. The density matrix estimation and the testing procedure are implemented and their numerical performances are studied.En optique quantique, la reconstruction de l'état quantique (fonction de Wigner ou matrice de densité infini-dimensionnelle) d'un faisceau de lumière correspond en statistique à un problème inverse trés mal posé. Premièrement, nous proposons des estimateurs de la matrice de densité basés sur les fonctions \textit{pattern} et des estimateurs à noyau de la fonction de Wigner. Nous faisons l'hypothèse que la matrice de densité inconnue appartient à une classe non paramétrique définie en accord avec les exemples étudiés par les physiciens. Nous en déduisons pour la fonction de Wigner associée à cette matrice des propriétés de décroissance rapide et de régularité. Deuxièmement, nous estimons une fonctionnelle quadratique de la fonction de Wigner par une U-statistique d'ordre deux sur une classe plus large. Cette fonctionnelle peut être vue comme une indication sur la pureté de l'état quantique considéré. Nous en déduisons un estimateur adaptatif aux paramètres de régularité de la fonction de Wigner. La dernière partie de ce manuscrit est consacrée au problème de test d'adéquation à la matrice de densité. Cette procédure est construite à partir d'un estimateur de type projection sur les fonctions \textit{pattern}. Nous étudions les bornes supérieures de type minimax de toutes ces procédures. Les procédures d'estimation de la matrice de densité et de test d'adéquation à une matrice de densité sont implémentées et leurs performances numériques sont étudiées

Nonparametric estimation in quantum homodyne tomography

PARIS7-Bibliothèque centrale (751132105) / SudocSudocFranceF

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

International audienc

State estimation in quantum homodyne tomography with noisy data

International audienceIn the framework of noisy quantum homodyne tomography with efficiency parameter $0 0$ and $