# Mutually unbiased bases in dimension six containing a product-vector basis

## TL;DR

This paper explores the structure of four mutually unbiased bases in six-dimensional complex space, focusing on the number of product vectors they contain, and constructs specific cases with limited product vectors.

## Contribution

It provides bounds on the number of product vectors in MUBs in dimension six and constructs an exceptional case with specific product vector counts.

## Key findings

- Most MUBs contain at most two product vectors.
- An exceptional case with three, two, and two product vectors is constructed.
- The study advances understanding of MUB structure in dimension six.

## Abstract

Excluding the existence of four MUBs in $\bbC^6$ is an open problem in quantum information. We investigate the number of product vectors in the set of four mutually unbiased bases (MUBs) in dimension six, by assuming that the set exists and contains a product-vector basis. We show that in most cases the number of product vectors in each of the remaining three MUBs is at most two. We further construct the exceptional case in which the three MUBs respectively contain at most three, two and two product vectors. We also investigate the number of vectors mutually unbiased to an orthonormal basis.