The H2-reducible matrix in four six-dimensional mutually unbiased bases

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 × 6 complex Hadamard matrix (CHM)...

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Bibliographic Details
Published in:Quantum information processing 2019-11, Vol.18 (11)
Main Authors: Liang, Mengfan, Hu, Mengyao, Chen, Lin, Chen, Xiaoyu
Format: Article
Language:English
Subjects:
Data Structures and Information Theory
Mathematical Physics
Physics
Physics and Astronomy
Quantum Computing
Quantum Information Technology
Quantum Physics
Spintronics
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Summary: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 × 6 complex Hadamard matrix (CHM) containing a 2 × 2 subunitary matrix. We show that every 6 × 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 × 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 × 2 subunitary matrices.
ISSN:1570-0755
1573-1332
DOI:10.1007/s11128-019-2467-3