Mutually unbiased bases in dimension six: The four most distant bases

TL;DR

This paper investigates the existence of four mutually unbiased bases in six-dimensional space, providing numerical evidence that such a set likely does not exist and exploring the structure of near-optimal bases.

Contribution

It offers the first numerical evidence that four mutually unbiased bases cannot exist in dimension six and introduces a family of three bases close to the maximum average distance.

Findings

Maximum average distance is less than unity, indicating no four mutually unbiased bases in dimension six.

A two-parameter family of three bases reaches the maximum average distance with the canonical basis.

Detailed structural analysis of extremal bases set is provided.

Abstract

We consider the average distance between four bases in dimension six. The distance between two orthonormal bases vanishes when the bases are the same, and the distance reaches its maximal value of unity when the bases are unbiased. We perform a numerical search for the maximum average distance and find it to be strictly smaller than unity. This is strong evidence that no four mutually unbiased bases exist in dimension six. We also provide a two-parameter family of three bases which, together with the canonical basis, reach the numerically-found maximum of the average distance, and we conduct a detailed study of the structure of the extremal set of bases.