  The aim of this work is to estimate a quadratic functional of a unknown
Wigner function from noisy tomographic data. The Wigner function can be seen as
the representation of the quantum state of a light beam. The estimation of a
quadratic functional is done from result of quantum homodyne measurement
performed on identically prepared quantum systems. We start by constructing an
estimator of a quadratic functional of the Wigner function. We show that the
proposed estimator is optimal or nearly optimal in a minimax sense over a class
of infinitely differentiable functions. Parametric rates are also reached for
some values of the smoothness parameters and the asymptotic normality is given.
Then, we construct an adaptive estimator that does not depend on the smoothness
parameters and prove it is minimax over some set-ups.
