Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data

Abstract

In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2<\eta \le 1$, we propose a novel estimator of a quantum state whose density matrix elements ${\rho }_{m,n}$ decrease like $C{e}^{-B(m+n{)}^{r/2}}$, for fixed $C\ge 1$, $B>0$ and $0<r\le 2$. On the contrary 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 conversgence as the best possible procedure relying on the knowledge of $(r,B,C)$. Finite sample behaviour of our adaptive procedure are explored through numerical experiments.

References in corpus (2)

• Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors
• Nonparametric goodness-of fit testing in quantum homodyne tomography with noisy data