Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

0544707 - ÚJF 2022 RIV PL eng J - Journal Article
Contreras, D. U. - Goyeneche, D. - Turek, Ondřej - Václavíková, Z.
Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1.
Communications in mathematics. Roč. 29, č. 1 (2021), s. 15-34. ISSN 1804-1388
Institutional support: RVO:61389005
Keywords : Circulant matrix * Hadamard matrix * Matually unbiased base * Orthogonal matrix
OECD category: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Method of publishing: Open access
https://doi.org/10.2478/cm-2021-0005

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 Zm. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.
Permanent Link: http://hdl.handle.net/11104/0321532