$H_2$-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.

arXiv comment15 pages, be published on quantum information processing. arXiv admin note: text overlap with arXiv:1904.10181

References in corpus (14)

• On mutually unbiased bases
• Constructing Mutually Unbiased Bases in Dimension Six
• Entanglement in mutually unbiased bases
• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• Quantifying measurement incompatibility of mutually unbiased bases
• A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
• Mutually unbiased bases in dimension six: The four most distant bases
• The Limited Role of Mutually Unbiased Product Bases in Dimension Six
• On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six
• Mutually Unbiased Product Bases for Multiple Qudits
• The Birkhoff theorem for unitary matrices of arbitrary dimensions
• Mutually unbiased bases in dimension six containing a product-vector basis
• Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras
• On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six