Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data

The aim of this paper is to answer an important issue in quantum me-chanics, namely to determine if a quantum state of a light beam is pure or mixed. The estimation of the purity is done from measurements by Quantum Homodyne Tomography performed on identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner func-tion, a density of generalized joint probability which can take negative values and which must respect certain constraints of positivity imposed by quantum physics. We propose to estimate a quadratic functional of the Wigner function by a kernel method as the physical measure of the purity of the state. We give also an adaptive estimator that does not depend on the smoothness parame-ters and we establish upper bound on the minimax risk over a class of infinitely differentiable functions. 1 ha l-