Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data ## AbstractWe 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 density on $\mathbb{R}^2$ which may take negative values but 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 and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions. We construct adaptive estimators, i.e. which do not depend on the smoothness parameters, and prove that in some set-ups they attain the minimax rates for the corresponding smoothness class.
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