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The \(H_2\)-reducible matrix in four six-dimensional mutually unbiased bases

Abstract

Finding four six-dimensional mutually unbiased bases (MUBs) is a long-standing open problem in quantum information. By assuming that they exist and contain the identity matrix, we investigate whether the remaining three MUBs have an \(H_2\)-reducible matrix, namely a \(6\times 6\) complex Hadamard matrix (CHM) containing a \(2\times 2\) subunitary matrix. We show that every \(6\times 6\) CHM containing at least 23 real entries is an \(H_2\)-reducible matrix. It relies on the fact that the CHM is complex equivalent to one of the two constant \(H_2\)-reducible matrices. They, respectively, have exactly 24 and 30 real entries, and both have more than eighteen \(2\times 2\) subunitary matrices. It turns out that such \(H_2\)- reducible matrices do not belong to the remaining three MUBs. This is the corollary of a stronger claim; namely, any \(H_2\)-reducible matrix belonging to the remaining three MUBs has exactly nine or eighteen \(2\times 2\) subunitary matrices.

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Notes

  1. Note that the standard definition should be \(H_nH_n^\dagger =I_n\), and we have added the coefficient n so that the entry \(u_{ij}\) has modulus one for simplicity. It does not influence the determination of CHM in an MUB trio.

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Acknowledgements

Authors were supported by the NNSF of China (Grant No. 11871089) and the Fundamental Research Funds for the Central Universities (Grant Nos. KG12080401 and ZG216S1902).

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Correspondence to Mengyao Hu or Lin Chen.

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Liang, M., Hu, M., Chen, L. et al. The \(H_2\)-reducible matrix in four six-dimensional mutually unbiased bases. Quantum Inf Process 18, 352 (2019). https://doi.org/10.1007/s11128-019-2467-3

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Keywords

  • Mutually unbiased basis
  • Six dimensions
  • Complex Hadamard matrix