On the Existence of Algebraic Equiangular Lines

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 [Formula: see text] equiangular unit vectors in [Formula: see text], then there must exist a set of [Formula: see text] 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, mutually unbiased bases, and consequences of the above results.