On the Existence of Algebraic Equiangular Lines

TL;DR

This paper investigates the existence and structure of maximal sets of equiangular lines in real and complex spaces, with implications for quantum measurement theory and algebraic number fields.

Contribution

It proves that maximal complex equiangular line sets with $d^2$ vectors can be chosen with coefficients in a number field, advancing understanding of SIC-POVMs.

Findings

Existence of algebraic structure in maximal complex equiangular lines

Applications to real equiangular lines

Implications for quantum measurement designs

Abstract

We consider real and complex equiangular lines, generated by unit vectors. We show that, for an arbitrary dimension $d$, if there exists a set of $d^2$ equiangular unit vectors in $\mathbb{C}^d$, then there must exist a set of $d^2$ equiangular unit vectors with all of their coefficients in a number field. This result is motivated by the question of constructing SIC-POVMs in quantum physics and conjectures around them. We discuss applications of our techniques to the case of real equiangular lines and consequences of the above results.