Generation of mutually unbiased bases as powers of a unitary matrix in 2-power dimensions

TL;DR

This paper constructs specific unitary matrices of order q+1 in 2-power dimensions that generate mutually unbiased bases, linking their existence to Lie algebra decompositions, advancing quantum information theory.

Contribution

It demonstrates the existence of unitary matrices of order q+1 generating mutually unbiased bases in 2-power dimensions using representation theory.

Findings

Existence of q x q unitary matrices of order q+1 for q a power of 2.

Construction of mutually unbiased bases from these matrices.

Connection between these matrices and orthogonal decompositions of Lie algebras.

Abstract

Let q be a power of 2. We show by representation theory that there exists a q x q unitary matrix of multiplicative order q+1 whose powers generate q+1 pairwise mutually unbiased base in C^q. When q is a power of an odd prime, there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the existence of these matrices implies the existence of a special type of orthogonal decomposition with respect to the Killing form of the special linear and symplectic Lie algebras.