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Article Dans Une Revue Communications in Mathematics Année : 2021

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

Daniel Uzcátegui Contreras
  • Fonction : Auteur correspondant
Dardo Goyeneche
  • Fonction : Auteur
Ondřej Turek
  • Fonction : Auteur
Zuzana Václavíková
  • Fonction : Auteur

Résumé

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.
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Dates et versions

hal-03665011 , version 1 (11-05-2022)

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Paternité - Pas d'utilisation commerciale - Pas de modification

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Daniel Uzcátegui Contreras, Dardo Goyeneche, Ondřej Turek, Zuzana Václavíková. Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1. Communications in Mathematics, 2021, Volume 29 (2021), Issue 1 (Special Issue: Ostrava Mathematical Seminar) (1), pp.15 - 34. ⟨10.2478/cm-2021-0005⟩. ⟨hal-03665011⟩
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