SIC-POVMs from Stark Units: Dimensions n^2+3=4p, p prime

TL;DR

This paper explores a number-theoretical approach to constructing symmetric informationally complete positive operator-valued measures (SIC-POVMs) in specific dimensions related to prime numbers, using Stark units from ray class fields.

Contribution

It extends a conjectural algorithm for SIC construction to dimensions of the form 4p, with p prime, and provides explicit solutions for seventeen such dimensions, improving understanding of the number-theoretical structure involved.

Findings

Solutions for 17 dimensions up to 39604.

The algorithm is easier to explain for even dimensions of the form 4p.

Partial improvements over previous methods are reported.

Abstract

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 (arXiv:2112.05552) we gave a conjectural algorithm for how to construct a SIC if $d = n^2+3 = p$, 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 $d = n^2+3$. Here we focus on the case when $d= n^2+3 = 4p$, $p$ 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 seventeen different dimensions of this form, reaching $d = 39604$. 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.