{"title":"The non-\\(H_2\\)-reducible matrices and some special complex Hadamard matrices","authors":"Mengfan Liang, Lin Chen, Fengyue Long, Xinyu Qiu","doi":"10.1007/s11128-024-04482-y","DOIUrl":null,"url":null,"abstract":"<div><p>Characterizing the <span>\\(6\\times 6\\)</span> complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the <span>\\(6\\times 6\\)</span> CHMs except the <span>\\(H_2\\)</span>-reducible matrices and the Tao matrix as the non-<span>\\(H_2\\)</span>-reducible matrices. As far as we know, no non-<span>\\(H_2\\)</span>-reducible matrices with analytic form have been found. In this paper, we find some non-<span>\\(H_2\\)</span>-reducible matrices with analytic form. We also characterize some special <span>\\(6\\times 6\\)</span> CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in <span>\\(\\mathbb {C}^6\\)</span> consists of an MUB trio and the identity). Our results may lead to the complete classification of <span>\\(6\\times 6\\)</span> CHMs and the solution of MUB problem in <span>\\(\\mathbb {C}^6\\)</span>.</p></div>","PeriodicalId":746,"journal":{"name":"Quantum Information Processing","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quantum Information Processing","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11128-024-04482-y","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
Characterizing the \(6\times 6\) complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the \(6\times 6\) CHMs except the \(H_2\)-reducible matrices and the Tao matrix as the non-\(H_2\)-reducible matrices. As far as we know, no non-\(H_2\)-reducible matrices with analytic form have been found. In this paper, we find some non-\(H_2\)-reducible matrices with analytic form. We also characterize some special \(6\times 6\) CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in \(\mathbb {C}^6\) consists of an MUB trio and the identity). Our results may lead to the complete classification of \(6\times 6\) CHMs and the solution of MUB problem in \(\mathbb {C}^6\).
期刊介绍:
Quantum Information Processing is a high-impact, international journal publishing cutting-edge experimental and theoretical research in all areas of Quantum Information Science. Topics of interest include quantum cryptography and communications, entanglement and discord, quantum algorithms, quantum error correction and fault tolerance, quantum computer science, quantum imaging and sensing, and experimental platforms for quantum information. Quantum Information Processing supports and inspires research by providing a comprehensive peer review process, and broadcasting high quality results in a range of formats. These include original papers, letters, broadly focused perspectives, comprehensive review articles, book reviews, and special topical issues. The journal is particularly interested in papers detailing and demonstrating quantum information protocols for cryptography, communications, computation, and sensing.