On the mathematical foundations of mutually unbiased bases

TL;DR

This paper explores the mathematical foundations of mutually unbiased bases (MUBs), focusing on Zauner's conjecture and its implications for the existence and construction of maximal MUB sets in complex spaces.

Contribution

It formulates fundamental questions about MUBs that impact the understanding and development of maximal MUB sets, especially in relation to Zauner's conjecture.

Findings

Identifies key questions related to the existence of maximal MUBs in non-prime power dimensions.

Highlights the importance of these questions for constructing new sets of MUBs.

Provides a theoretical framework for future research on MUBs.

Abstract

In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in $\mathbb{C}^d$, a set of MUBs of the theoretical maximal size $d + 1$ exists only if $d$ is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.