The observational limit of wave packets with noisy measurements

TL;DR

This paper investigates the limits of using wave packets to recover observables from noisy measurements, highlighting the potential and limitations of this approach in the presence of Gaussian noise.

Contribution

The paper introduces a method using wave packets for observable recovery from noisy data and analyzes its limitations due to noise-induced signal hiding.

Findings

Wave packets enable partial recovery of the observable.

Gaussian noise limits the complete recovery of the observable.

Ergodicity of errors underpins the recovery process.

Abstract

The authors consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian white noise. The authors show how wave packets can be used to partially recover the observable from the measurements almost surely. Furthermore, they point out the limitation of wave packets to recover the remaining part of the observable, and show how the errors hide the signal coming from the observable. The recovery results are based on an ergodicity property of the errors produced by wave packets.