Santiago Barrera Acevedo, Heiko Dietrich, Corey Lionis
{"title":"新的四元哈达玛矩阵族","authors":"Santiago Barrera Acevedo, Heiko Dietrich, Corey Lionis","doi":"10.1007/s10623-024-01401-1","DOIUrl":null,"url":null,"abstract":"<p>A quaternionic Hadamard matrix (QHM) of order <i>n</i> is an <span>\\(n\\times n\\)</span> matrix <i>H</i> with non-zero entries in the quaternions such that <span>\\(HH^*=nI_n\\)</span>, where <span>\\(I_n\\)</span> and <span>\\(H^*\\)</span> denote the identity matrix and the conjugate-transpose of <i>H</i>, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New families of quaternionic Hadamard matrices\",\"authors\":\"Santiago Barrera Acevedo, Heiko Dietrich, Corey Lionis\",\"doi\":\"10.1007/s10623-024-01401-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A quaternionic Hadamard matrix (QHM) of order <i>n</i> is an <span>\\\\(n\\\\times n\\\\)</span> matrix <i>H</i> with non-zero entries in the quaternions such that <span>\\\\(HH^*=nI_n\\\\)</span>, where <span>\\\\(I_n\\\\)</span> and <span>\\\\(H^*\\\\)</span> denote the identity matrix and the conjugate-transpose of <i>H</i>, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-06-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10623-024-01401-1\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01401-1","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
阶数为 n 的四元哈达玛矩阵（QHM）是一个在四元中具有非零条目的 \(n/times n\) 矩阵 H，使得 \(HH^*=nI_n\) ，其中 \(I_n\) 和 \(H^*\) 分别表示 H 的同位矩阵和共轭变换。如果一个 QHM 的第一行和第一列的所有条目都是 1，那么它就是去相的，如果它的条目产生一个非交换群，那么它就是非交换的。我们工作的目的是提供无限多（非交换去相位）QHM 的新构造；Farkas 等人（IEEE Trans Inform Theory 69(6):3814-3824, 2023）利用这些矩阵产生了互不偏倚的测量结果。
A quaternionic Hadamard matrix (QHM) of order n is an \(n\times n\) matrix H with non-zero entries in the quaternions such that \(HH^*=nI_n\), where \(I_n\) and \(H^*\) denote the identity matrix and the conjugate-transpose of H, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
