An application of positive definite functions to the problem of MUBs

TL;DR

This paper introduces a novel approach using positive definite functions on the unitary group to analyze mutually unbiased bases (MUBs), providing new proofs and potential insights into their maximal number and existence in specific dimensions.

Contribution

It offers a new proof for the maximum number of MUBs in complex dimensions and suggests a method that could prove the non-existence of complete MUB systems in dimension 6.

Findings

Proof that there are at most d+1 MUBs in C^d

New approach based on positive definite functions

Potential proof of non-existence of complete MUBs in dimension 6

Abstract

We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most $d+1$ MUBs in ${\mathbb C}^d$. It may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.