$H_2$-reducible matrices in six-dimensional mutually unbiased bases

Abstract

Finding four six-dimensional mutually unbiased bases (MUBs) containing the identity matrix is a long-standing open problem in quantum information. We show that if they exist, then the $H_2$-reducible matrix in the four MUBs has exactly nine $2\times 2$ Hadamard submatrices. We apply our result to exclude from the four MUBs some known CHMs, such as symmetric $H_2$-reducible matrix, the Hermitian matrix, Dita family, Bjorck's circulant matrix, and Szollosi family. Our results represent the latest progress on the existence of six-dimensional MUBs.

15 pages, be published on quantum information processing. arXiv admin note: text overlap with arXiv:1904.10181

References in corpus (4)

• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six
• The Birkhoff theorem for unitary matrices of arbitrary dimensions
• Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras