On the Existence of Algebraic Equiangular Lines

• URL: http://arxiv.org/abs/2603.09128v1
• Date: Tue, 10 Mar 2026 03:01:34 GMT
• Title: On the Existence of Algebraic Equiangular Lines
• Abstract summary: We consider real and complex equiangular lines, generated by unit vectors.<n>We show that, for an arbitrary dimension $d$, if there exists a set of $d2$ equiangular unit vectors in $mathbbCd$, then there must exist a set of $d2$ equiangular unit vectors with all of their coefficients in a number field.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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