Statistique pour la physique quantique – StatQuant

Thanks to recent technological progress, the realization of a wide range of quantum states is now possible, and paves the way to the actual implementation of the quantum information technology that has been theoretically developped in the last decade. Measuring a quantum state from observations that are necessarily random is, by definition, a statistical problem. Physical aspects have been developped in numerous European laboratories, and the statistical problems have recently gained in importance. Generally speaking, a quantum state is represented by a self-adjoint operator ho on a Hilbert state; an observable is another operator M = sum_a M_a, where M_a is the projector on the eigenspace associated to the eigenvalue a. Measuring this observable gives as a result a with probability = trace( ho M_a). Quantum statistics consists in reconstructing ho from a sequence of such measures. In discrete variables (finite-dimensional Hilbert space), the main interest and difficulty come from the fact that the system made of several quantum systems considered simultaneously has much more possible states than the same systems considered separately. Simultaneous measure gives more information, but it is more difficult to realize physically and to analyze statistically. In continous variables, the quantum state of light in a cavity can be represented by the Wigner distribution in two variables (the distribution of the "electric field" and "magnetic field" observables can then be recovered as marginal laws). The technique called quantum homodyne tomography allows to reconstruct this distribution by solving an inverse problem: this is a more classical statistics problem, but on which much work remains to do. New techniques allow to measure the Wigner distribution directly, but their cost makes it necessary to develop a non-parametric experiment planning and a sequential analysis to optimize the estimation. We propose in this project to develop these aspects: on one hand by guiding the experimental design to optimize the measures; on the other hand by designing new parametric (in finite dimension) and non-parametric (for the Wigner distribution) estimators and tests, maximizing their efficiency according to various critieria (the L^1 norm, in particular, poses specific problems), and analyzing this efficiency in a finer way (local asymptotic normality). Specific attention will be devoted to breakthrough methods for systems observed in continuous time.