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

• URL: http://arxiv.org/abs/2110.13646v2
• Date: Thu, 28 Oct 2021 01:51:16 GMT
• Title: $H_2$-reducible matrices in six-dimensional mutually unbiased bases
• Authors: Xiaoyu Chen, Mengfan Liang, Mengyao Hu, Lin Chen
• Abstract summary: 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 $H$-reducible matrix in the four MUBs has exactly nine $2times2$ Hadamard submatrices. Our results represent the latest progress on the existence of six-dimensional MUBs.
• Score: 9.320472997908771
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 $2\times2$ 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.

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