Product states and Schmidt rank of mutually unbiased bases in dimension six

TL;DR

This paper investigates the structure of mutually unbiased bases in six-dimensional complex space, revealing limitations on product states and Schmidt rank, and explores the connection between Sinkhorn normal form and unbiased vectors.

Contribution

It establishes bounds on product states and Schmidt rank in MUBs in dimension six, and links Sinkhorn normal form to the existence of unbiased vectors.

Findings

Any basis in a set of four MUBs in dimension six contains at most two product states.

Such bases have Schmidt rank at least three.

The Sinkhorn normal form relates to the existence of vectors unbiased to two bases.

Abstract

We show that if a set of four mutually unbiased bases (MUBs) in $\mathbb{C}^6$ exists and contains the identity, then any other basis in the set contains at most two product states and at the same time has Schmidt rank at least three. Here both the product states and the Schmidt rank are defined over the bipartite space $\mathbb{C}^2\otimes\mathbb{C}^3$. We also investigate the connection of the Sinkhorn normal form of unitary matrices to the fact that there is at least one vector unbiased to any two orthonormal bases in any dimension.