Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography
研究了在噪声量子零差层析成像中，用核估计器估计Wigner函数的理论性能，证明了其对于无穷可微函数类在L∞风险下达到极小极大最优（至多对数因子），并构造了不依赖光滑参数的自适应估计器。
In quantum optics, the quantum state of a light beam 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. In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2<\eta\leq1$, 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 $\mathbb{L}_{\infty}$-risk over a class of infinitely differentiable functions. We also compute the lower bound for the $\mathbb{L}_{2}$-risk. We construct an adaptive estimator, that is, which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness of the class of functions up to a logarithmic factor in the sample size. Finite sample behaviour of our adaptive procedure is explored through numerical experiments.