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

Abstract

Excluding the existence of four MUBs in ${\mathbb{C}}^6$ 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.

arXiv comment10 pages, the total number of vectors in the introduction and the conclusion was revised. arXiv admin note: text overlap with arXiv:1610.04875

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• Real Entries of Complex Hadamard Matrices and Mutually Unbiased Bases in Dimension Six
• Constructions of mutually unbiased entangled bases
• $H_2$-reducible matrices in six-dimensional mutually unbiased bases
• Some special complex Hadamard matrices of order six