Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1
| Autor: | Zuzana Václavíková, Ondřej Turek, Dardo Goyeneche, Daniel Uzcátegui Contreras |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
15b36
Root of unity General Mathematics Diagonal mutually unbiased base Absolute value (algebra) orthogonal matrix 15b10 Combinatorics Matrix (mathematics) QA1-939 FOS: Mathematics Mathematics - Combinatorics [MATH]Mathematics [math] Circulant matrix Mathematics circulant matrix 15b05 Computer Science::Information Retrieval Order (ring theory) Hermitian matrix hadamard matrix Combinatorics (math.CO) 15B10 15B36 15B05 Mutually unbiased bases |
| Zdroj: | Communications in Mathematics, Vol 29, Iss 1, Pp 15-34 (2021) |
| DOI: | 10.2478/cm-2021-0005 |
| Popis: | It is known that a real symmetric circulant matrix with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ 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 $d\geq0$ 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. Comment: 16 pages, revised version: text partly rewritten, several new results added |
| Databáze: | OpenAIRE |
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