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

TL;DR

This paper introduces an adaptive estimator for quantum state density matrices in noisy homodyne tomography, achieving optimal convergence rates without prior knowledge of decay parameters, validated through numerical experiments.

Contribution

It develops a novel adaptive estimation method that is rate-optimal without knowing decay parameters, advancing quantum state tomography techniques.

Findings

Estimator achieves rate-optimal convergence under $\, ext{L}_2$-loss.

Method adapts to unknown decay parameters $r$, $B$, and $C$.

Numerical experiments confirm finite sample effectiveness.

Abstract

In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < \eta \leq 1$, we propose a novel estimator of a quantum state whose density matrix elements $\rho_{m,n}$ decrease like $Ce^{-B(m+n)^{r/ 2}}$, for fixed $C\geq 1$, $B>0$ and $0<r\leq 2$. 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.