State estimation in quantum homodyne tomography with noisy data

TL;DR

This paper introduces two new estimators for quantum state reconstruction in noisy homodyne tomography, analyzing their convergence rates and effectiveness under different efficiency conditions.

Contribution

It proposes novel projection and kernel estimators for quantum states with specific decay properties, extending methods to lower efficiency regimes and providing convergence analysis.

Findings

Projection estimator for $0< ext{efficiency}\, ext{parameter}\,rac{1}{2}$

Kernel estimator for Wigner function reconstruction

Derived convergence rates in $\, ext{L}_2$ risk for both estimators

Abstract

In the framework of noisy quantum homodyne tomography with efficiency parameter $0 < \eta \leq 1$, we propose two estimators of a quantum state whose density matrix elements $\rho_{m,n}$ decrease like $e^{-B(m+n)^{r/ 2}}$, for fixed known $B>0$ and $0<r\leq 2$. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for $0<\eta \leq 1/2$), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in $\mathbb{L}_2$ risk.