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

Lin Chen, Li Yu

Published 2016-10-16Version 1

We show that if a set of four mutually unbiased bases (MUBs) in $\mathbb{C}^6$ 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 the Schmidt rank is defined over the bipartite space $\mathbb{C}^2\otimes\mathbb{C}^3$. We also present a simple argument for why there is at least one vector unbiased to any two given orthonormal bases in any dimension.