Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data

TL;DR

This paper develops and analyzes a kernel estimator for the Wigner function in noisy quantum homodyne tomography, demonstrating its minimax optimality and adaptability across smoothness classes, with supporting numerical experiments.

Contribution

It introduces an adaptive kernel estimator for the Wigner function that is minimax optimal in noisy quantum tomography, without requiring prior smoothness knowledge.

Findings

Estimator is minimax efficient up to a logarithmic factor.

Lower bounds for the $ ext{L}_2$-risk are established.

Numerical experiments confirm finite sample performance.

Abstract

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 $1/2 < \eta \leq 1$, 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 $\mathbb L_\infty$-risk over a class of infinitely differentiable. We compute also the lower bound for the $\mathbb L_2$-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.