The finite Fourier Transform and projective 2-designs

TL;DR

This paper revisits a unique eigenvector decomposition of the finite Fourier Transform in prime dimensions and demonstrates how applying Weyl-Heisenberg matrices generates a projective 2-design, linking Fourier analysis and quantum design theory.

Contribution

It introduces a novel connection between finite Fourier eigenvectors and projective 2-designs using Weyl-Heisenberg matrices in prime dimensions.

Findings

Eigenvector basis for finite Fourier Transform in prime dimensions

Application of Weyl-Heisenberg matrices generates a projective 2-design

Revisits and extends Balian and Itzykson's 1986 solution

Abstract

There are several approaches to define an eigenvector decomposition of the finite Fourier Transform, which is in some sense unique, and at best resembles the eigenstates of the quantum harmonic oscillator. A solution given by Balian and Itzykson in 1986 for prime dimensions d = 3 (mod 4) is revisited. It is shown, that by applying the Weyl-Heisenberg matrices to this eigenvector basis, a projective 2-design is generated.