Quadratic discrete Fourier transform and mutually unbiased bases

TL;DR

This paper introduces a quadratic discrete Fourier transform, extending the classical transform with quadratic parameters, and explores its application in generating mutually unbiased bases in quantum information, especially in prime dimensions.

Contribution

It presents the quadratic discrete Fourier transform as a novel extension of the classical Fourier transform and demonstrates its utility in quantum information for constructing mutually unbiased bases.

Findings

Quadratic discrete Fourier transform generalizes the classical transform.

In prime dimensions, it helps generate complete sets of mutually unbiased bases.

The transform reduces to the usual Fourier transform when parameters are zero.

Abstract

The present chapter [submitted for publication in "Fourier Transforms, Theory and Applications", G. Nikolic (Ed.), InTech (Open Access Publisher), Vienna, 2011] is concerned with the introduction and study of a quadratic discrete Fourier transform. This Fourier transform can be considered as a two-parameter extension, with a quadratic term, of the usual discrete Fourier transform. In the case where the two parameters are taken to be equal to zero, the quadratic discrete Fourier transform is nothing but the usual discrete Fourier transform. The quantum quadratic discrete Fourier transform plays an important role in the field of quantum information. In particular, such a transformation in prime dimension can be used for obtaining a complete set of mutually unbiased bases.