  We estimate the quantum state of a light beam from results of quantum
homodyne measurements performed on identically prepared quantum systems. The
state is represented through the Wigner function, a generalized probability
density on $\mathbb{R}^2$ which may take negative values and must respect
intrinsic positivity constraints imposed by quantum physics. The effect of the
losses due to detection inefficiencies, which are always present in a real
experiment, is the addition to the tomographic data of independent Gaussian
noise. We construct a kernel estimator for the Wigner function, prove that it
is minimax efficient for the pointwise risk over a class of infinitely
differentiable functions, and implement it for numerical results. We construct
adaptive estimators, that is, which do not depend on the smoothness parameters,
and prove that in some setups they attain the minimax rates for the
corresponding smoothness class.
