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arXiv:2603.09128 · Journal of Algebra and Its Applications

On the Existence of Algebraic Equiangular Lines

Igor Van Loo, Frédérique Oggier

quant-phmath.NTSubmitted Mar 10, 2026• Announced Mar 11, 2026• 0 citations
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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.

DOI
10.1142/S0219498828500508
Comments
21 pages, 1 Table

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We consider real and complex equiangular lines, generated by unit vectors. We show that, for an arbitrary dimension ddd, if there exists a set of d2d^2d2 equiangular unit vectors in Cd\mathbb{C}^dCd, then there must exist a set of d2d^2d2 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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Abstract The existence of d2d^2d2 pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in ddd-dimensional Hilbert space is known only for finitely many dimensions ddd. We prove that, if there exists a set of real units in a certain ray class field (depending on ddd) satisfying certain algebraic properties, a SIC-POVM exists, when ddd is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0s=0s=0 and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.
The existence of pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in -dimensional Hilbert space is known only for finitely many dimensions ⁠. We prove that, if there exists a set of real units in a certain ray class field (depending on ⁠) satisfying certain algebraic properties, a SIC-POVM exists, when is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.
A set of vectors of equal norm in Cd\mathbb{C}^dCd represents equiangular lines if the magnitudes of the Hermitian inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is d2d^2d2, and it is conjectured that sets of this maximum size exist in Cd\mathbb{C}^dCd for every d≥2d \geq 2d≥2. We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following 3 constructions of equiangular lines: (1) adapting a set of ddd MUBs in Cd\mathbb{C}^dCd to obtain d2d^2d2 equiangular lines in Cd\mathbb{C}^dCd, (2) using a set of ddd MUBs in Cd\mathbb{C}^dCd to build (2d)2(2d)^2(2d)2 equiangular lines in C2d\mathbb{C}^{2d}C2d, (3) combining two copies of a set of ddd MUBs in Cd\mathbb{C}^dCd to build (2d)2(2d)^2(2d)2 equiangular lines in C2d\mathbb{C}^{2d}C2d. For each construction, we give the dimensions ddd for which we currently know that the construction produces a maximum-sized set of equiangular lines.
The existence problem for maximal sets of equiangular lines (or SICs) in complex Hilbert space of dimension d remains largely open. In a previous publication we gave a conjectural algorithm for how to construct a SIC if d = n2 + 3 = p, a prime number. Perhaps the most surprising number-theoretical aspect of that algorithm is the appearance of Stark units in a key role: a single Stark unit from a ray class field extension of a real quadratic field serves as a seed from which the SIC is constructed. The algorithm can be modified to apply to all dimensions d = n2 + 3. Here we focus on the case when d = n2 + 3 = 4p, p prime, for two reasons. First, special measures have to be taken on the Hilbert space side of the problem when the dimension is even. Second, the degrees of the relevant ray class fields are “smooth” in a sense that facilitates exact calculations. As a result the algorithm becomes easier to explain. We give solutions for 17 different dimensions of this form, reaching d = 39 604. Several improvements relative to our previous publication are reported, but we cannot offer a proof that the algorithm works for any dimensions where it has not been tested.
A well supported conjecture states that SIC-POVMs -- maximal sets of complex equiangular lines -- with anti-unitary symmetry give rise to an identity expressing some of its overlaps as squares of the (rescaled) components of a suitably chosen fiducial vector. In number theoretical terms the identity essentially expresses Stark units as sums of products of pairs of square roots of Stark units. We investigate whether the identity is enough to determine these Stark units. The answer is no, but the failure might be quite mild.