| Title:
|
Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$ (English) |
| Author:
|
Contreras, Daniel Uzcátegui |
| Author:
|
Goyeneche, Dardo |
| Author:
|
Turek, Ondřej |
| Author:
|
Václavíková, Zuzana |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
29 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
15-34 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is known that a real symmetric circulant matrix with diagonal entries $d\geq 0$, off-diagonal entries $\pm 1$ and orthogonal rows exists only of order $2d+2$ (and trivially of order $1$) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d\geq 0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with $d$ different from an odd integer is $n=2d+2$. We also discuss a similar problem for symmetric circulant matrices defined over finite rings $\mathbb {Z}_m$. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory. (English) |
| Keyword:
|
Circulant matrix |
| Keyword:
|
orthogonal matrix |
| Keyword:
|
Hadamard matrix |
| Keyword:
|
mutually unbiased base |
| MSC:
|
15B05 |
| MSC:
|
15B10 |
| MSC:
|
15B36 |
| idZBL:
|
Zbl 07413355 |
| idMR:
|
MR4251308 |
| . |
| Date available:
|
2021-07-09T12:23:21Z |
| Last updated:
|
2021-11-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148989 |
| . |
| Reference:
|
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