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

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.

23 pages. Removed some unneccesary lemmas, and revised the introduction

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Cited by in corpus (3)

• Comment on 'Product states and Schmidt rank of mutually unbiased bases in dimension six'
• Reply to Comment on 'Product states and Schmidt rank of mutually unbiased bases in dimension six'
• Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three