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Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data
Karim Lounici, Katia Meziani, Gabriel Peyré
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In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on ℝ² which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2 < η≤1, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the 𝕃_∞-risk over a class of infinitely differentiable. We compute also the lower bound for the 𝕃₂-risk. We construct adaptive estimator, i.e. which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness class functions. Finite sample behaviour of our adaptive procedure are explored through numerical experiments.
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