An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

TL;DR

This paper introduces a novel approach combining angular momentum theory and SU(2) representation theory to construct unitary operator bases, linking concepts like Fourier transforms, Hadamard matrices, and Gauss sums in finite-dimensional quantum systems.

Contribution

It provides a unified formula for constructing unitary bases using polar decomposition of SU(2), connecting various mathematical structures in quantum information theory.

Findings

Derived a single formula for unitary bases from SU(2) polar decomposition

Linked quadratic Fourier transforms, Hadamard matrices, and Gauss sums in a unified framework

Explored applications to Weyl pairs, Pauli operators, and the unitary group

Abstract

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.