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\(H_2\)-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.

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Acknowledgements

XYC and MFL were partially supported by NSFC (Grant No. 61702025) and State Key Laboratory of Software Development Environment (Grant No. SKLSDE-2019ZX-12). MYH and LC were supported by the NNSF of China (Grant No. 11871089), and the Fundamental Research Funds for the Central Universities (Grant No. ZG216S1902).

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Correspondence to Mengfan Liang, Mengyao Hu or Lin Chen.

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Chen, X., Liang, M., Hu, M. et al. \(H_2\)-reducible matrices in six-dimensional mutually unbiased bases. Quantum Inf Process 20, 353 (2021). https://doi.org/10.1007/s11128-021-03278-8

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  • DOI: https://doi.org/10.1007/s11128-021-03278-8

Keywords

  • Complex Hadamard matrix
  • \(H_2\)-Reducible matrix
  • Mutually unbiased bases

Mathematics Subject Classification

  • 15A21
  • 15A51