Daniel Qin: On Mutually Unbiased Bases and Hadamard Matrices

Abstract

The problem of finding maximal sets of mutually unbiased bases in arbitrary dimensions is fairly young despite being related to a wide range of "older'' combinatorial objects such finite projective planes. Of these equivalent objects, our discussion will focus on the standard constructions of mutually unbiased bases by unitary operators and complex Hadamard matrices. We aim to understand why these existing constructions work in prime power dimensions and fail in non-prime power dimensions—in particular, in the first composite dimension, six. We will wrap up by working a few explicit computations and suggesting some other methods to explore to progress along the existence problem.