A Fourier analytic approach to the problem of mutually unbiased bases

TL;DR

This paper introduces a novel Fourier analytic method in additive combinatorics to study mutually unbiased bases (MUBs), offering a concise proof of their maximum number in complex dimensions and potential insights into their existence in composite dimensions.

Contribution

It presents a new Fourier analytic approach to MUBs, generalizing existing bounds and potentially resolving the open problem of their existence in certain dimensions.

Findings

Provides a short proof that there are at most d+1 MUBs in 

Introduces a Fourier analytic technique for MUBs

Suggests a possible proof of non-existence of complete MUB systems in some composite dimensions

Abstract

We give an entirely new approach to the problem of mutually unbiased bases (MUBs), based on a Fourier analytic technique in additive combinatorics. The method provides a short and elegant generalization of the fact that there are at most $d+1$ MUBs in $\Co^d$. It may also yield a proof that no complete system of MUBs exists in some composite dimensions -- a long standing open problem.