On the Existence of Algebraic Equiangular Lines

Igor Van Loo; Frédérique Oggier

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arXiv · 2603.09128

On the Existence of Algebraic Equiangular Lines

Igor Van Loo·Frédérique Oggier

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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.

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Igor Van Loo, Frédérique Oggier. 2026-03-10. On the Existence of Algebraic Equiangular Lines. https://arxiv.org/abs/2603.09128

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