eng
episciences.org
Communications in Mathematics
1804-1388
2336-1298
2021-04-30
Volume 29 (2021), Issue 1...
10.2478/cm-2021-0005
9524
journal article
Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1
Daniel Uzcátegui Contreras
Dardo Goyeneche
Ondřej Turek
Zuzana Václavíková
It is known that a real symmetric circulant matrix with diagonal entries d ≥ 0, off-diagonal entries ±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 ≥ 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 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.
https://cm.episciences.org/9524/pdf
General Mathematics
[MATH]Mathematics [math]