Gaussian Distributions and Phase Space Weyl--Heisenberg Frames

TL;DR

This paper develops explicit formulas and error estimates for approximating Gaussian symbols in phase space using phase space shifted standard Gaussians, advancing quantum information processing techniques.

Contribution

It introduces the concept of phase space frames associated with Weyl-Heisenberg frames and provides new approximation formulas for Gaussian symbols.

Findings

Explicit approximation formulas for Gaussian symbols.

Error estimates and asymptotic behavior analysis.

Application to quantum information processing.

Abstract

Gaussian states are at the heart of quantum mechanics and play an essential role in quantum information processing. In this paper we provide approximation formulas for the expansion of a general Gaussian symbol in terms of elementary Gaussian functions. For this purpose we introduce the notion of a "phase space frame" associated with a Weyl-Heisenberg frame. Our results give explicit formulas for approximating general Gaussian symbols in phase space by phase space shifted standard Gaussians as well as explicit error estimates and the asymptotic behavior of the approximation.