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

TL;DR

This paper explores the existence and construction of complex Hermitian circulant matrices with orthogonal rows, diagonal entries, and off-diagonal entries of absolute value 1, revealing new insights and connections to quantum information theory.

Contribution

It introduces the complex Hermitian analogue of known real symmetric circulant matrices and establishes conditions for their existence, including cases with 4th roots of unity and finite rings.

Findings

Hermitian circulant matrices exist under specific order conditions

Order of matrices with 4th roots of unity off-diagonal entries is 2d+2 for non-odd d

Connections to mutually unbiased bases in quantum information

Abstract

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.