On Mutually Unbiased Bases and Hadamard Matrices : An exploration from quantum mechanics to number theory

Abstract

The goal of this paper will be to study the behavior of mutually unbiased bases in complex dimensions by looking at equivalent objects that preserve the mutually unbiasedness property. Of the many equivalent objects, this paper will focus on the unitary operators and complex Hadamard matrices. We will thoroughly examine the standard constructions of mutually unbiased bases using these two approaches. By understanding why the existing constructions work in prime power dimensions (and fail in non-prime dimensions) we aim to motivate new methods to tackle their existence in non-prime power dimension{in particular, in the first composite dimension, six.