---
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.'
author:
- |
    J M Aubry$^1$, C Butucea$^2$ and K Meziani$^3$\
    $^1$\
    \
    $^2$\
    \
    $^3$\
    \
bibliography:
- 'Wignerbib.bib'
title: State estimation in quantum homodyne tomography with noisy data
---

**Keywords**: density matrix, Gaussian noise, $\mathbb{L}_2$-risk, nonparametric estimation, pattern functions, quantum homodyne tomography, quantum state, Radon transform, Wigner function.\
**AMS 2000 subject classifications:** [62G05, 62G20, 81V80]{}\

Introduction
============

Experiments in quantum optics consist in creating, manipulating and measuring quantum states of light. The technique called quantum homodyne tomography allows to retrieve partial, noisy information from which the state is to be recovered: this is the subject of the present chapter.

Quantum states
--------------

Mathematically, the main concepts of quantum mechanics 