Nonparametric goodness-of fit testing in quantum homodyne tomography with noisy data

TL;DR

This paper introduces a new goodness-of-fit testing method for quantum homodyne tomography with noisy data, addressing the challenges of ill-posed inverse problems in quantum optics.

Contribution

It presents a novel projection-based testing procedure with analyzed convergence rates under different smoothness conditions.

Findings

The method effectively detects deviations in quantum state data.

Convergence rates depend on the smoothness assumptions.

Simulations demonstrate the procedure's practical applicability.

Abstract

In the framework of quantum optics, we study the problem of goodness-of-fit testing in a severely ill-posed inverse problem. A novel testing procedure is introduced and its rates of convergence are investigated under various smoothness assumptions. The procedure is derived from a projection-type estimator, where the projection is done in $\mathbb{L}_2$ distance on some suitably chosen pattern functions. The proposed methodology is illustrated with simulated data sets.