arXiv:1610.04875 · Journal of Physics A Mathematical and Theoretical

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

Lin Chen, Li Yu

Published Sep 28, 2017• 19 citations

Abstract

Abstract We show that if a set of four mutually unbiased bases (MUBs) in 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 C 2 ⊗ 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.

DOI: 10.1088/1751-8121/aa8f9e

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Comment on ‘Product states and Schmidt rank of mutually unbiased bases in dimension six’

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Abstract A lemma by Chen et al (2017 J. Phys. A: Math. Theor. 50 475304 ) provides a necessary condition on the structure of any complex Hadamard matrix in a set of four mutually unbiased bases in C 6 . The proof of the lemma is shown to contain a mistake, ultimately invalidating three theorems derived in later publications.

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