Résumé en anglais

In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2 < η ≤ 1, we propose a novel estimator of a quantum state whose density matrix elements ρm, n decrease like $C{\rm e}^{-B(m+n)^{ r/ 2}}$, for fixed C ≥ 1, B > 0 and 0 < r ≤ 2. In contrast to previous works, we focus on the case where r, C and B are unknown. The procedure estimates the matrix coefficients by a projection method on the pattern functions, and then by soft-thresholding the estimated coefficients. We prove that under the $\mathbb {L}_2$-loss our procedure is adaptive rate-optimal, in the sense that it achieves the same rate of convergence as the best possible procedure relying on the knowledge of (r, B, C). Finite sample behaviour of our adaptive procedure is explored through numerical experiments.