The non-H2-reducible matrices and some special complex Hadamard matrices

Mengfan Liang; Lin Chen; Fengyue Long; Xinyu Qiu

doi:10.1007/s11128-024-04482-y

The non-H2-reducible matrices and some special complex Hadamard matrices

Mengfan Liang
, Lin Chen^*
, Fengyue Long
, Xinyu Qiu^*

^*Corresponding author for this work

School of Mathematical Sciences

Beihang University

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Characterizing the 6×6 complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the 6×6 CHMs except the H₂-reducible matrices and the Tao matrix as the non-H₂-reducible matrices. As far as we know, no non-H₂-reducible matrices with analytic form have been found. In this paper, we find some non-H₂-reducible matrices with analytic form. We also characterize some special 6×6 CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in C⁶ consists of an MUB trio and the identity). Our results may lead to the complete classification of 6×6 CHMs and the solution of MUB problem in C⁶.

Original language	English
Article number	278
Journal	Quantum Information Processing
Volume	23
Issue number	7
DOIs	https://doi.org/10.1007/s11128-024-04482-y
State	Published - Jul 2024

Keywords

15B34
Complex Hadamard matrix
Mutually unbiased bases
Non-H-reducible matrices

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10.1007/s11128-024-04482-y

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abstract = "Characterizing the 6×6 complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the 6×6 CHMs except the H2-reducible matrices and the Tao matrix as the non-H2-reducible matrices. As far as we know, no non-H2-reducible matrices with analytic form have been found. In this paper, we find some non-H2-reducible matrices with analytic form. We also characterize some special 6×6 CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in C6 consists of an MUB trio and the identity). Our results may lead to the complete classification of 6×6 CHMs and the solution of MUB problem in C6.",

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author = "Mengfan Liang and Lin Chen and Fengyue Long and Xinyu Qiu",

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T1 - The non-H2-reducible matrices and some special complex Hadamard matrices

AU - Liang, Mengfan

AU - Chen, Lin

AU - Long, Fengyue

AU - Qiu, Xinyu

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

PY - 2024/7

Y1 - 2024/7

N2 - Characterizing the 6×6 complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the 6×6 CHMs except the H2-reducible matrices and the Tao matrix as the non-H2-reducible matrices. As far as we know, no non-H2-reducible matrices with analytic form have been found. In this paper, we find some non-H2-reducible matrices with analytic form. We also characterize some special 6×6 CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in C6 consists of an MUB trio and the identity). Our results may lead to the complete classification of 6×6 CHMs and the solution of MUB problem in C6.

AB - Characterizing the 6×6 complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the 6×6 CHMs except the H2-reducible matrices and the Tao matrix as the non-H2-reducible matrices. As far as we know, no non-H2-reducible matrices with analytic form have been found. In this paper, we find some non-H2-reducible matrices with analytic form. We also characterize some special 6×6 CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in C6 consists of an MUB trio and the identity). Our results may lead to the complete classification of 6×6 CHMs and the solution of MUB problem in C6.

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KW - Complex Hadamard matrix

KW - Mutually unbiased bases

KW - Non-H-reducible matrices

UR - https://www.scopus.com/pages/publications/85199046936

U2 - 10.1007/s11128-024-04482-y

DO - 10.1007/s11128-024-04482-y

M3 - 文章

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SN - 1570-0755

VL - 23

JO - Quantum Information Processing

JF - Quantum Information Processing

IS - 7

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ER -

The non-H2-reducible matrices and some special complex Hadamard matrices

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