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 d² equiangular unit vectors in C^d, then there must exist a set of d² 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.