SIC-POVMs from Stark units: Prime dimensions n^2+3

Abstract

We propose a recipe for constructing a SIC fiducial vector in complex Hilbert space of dimension of the form $d=n^2+3$, focussing on prime dimensions $d=p$. Such structures are shown to exist in thirteen prime dimensions of this kind, the highest being $p=19603$. The real quadratic base field $K$ (in the standard SIC terminology) attached to such dimensions has fundamental units $u_K$ of norm $-1$. Let $\mathbb{Z}_K$ denote the ring of integers of $K$, then $p\mathbb{Z}_K$ splits into two ideals $\mathfrak{p}$ and $\mathfrak{p}'$. The initial entry of the fiducial is the square $\xi^2$ of a geometric scaling factor $\xi$, which lies in one of the fields $K(\sqrt{u_K})$. Strikingly, the other $p-1$ entries of the fiducial vector are each the product of $\xi$ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at $s=0$ of the first derivatives of partial $L$ functions attached to the characters of the ray class group of $\mathbb{Z}_K$ with modulus $\mathfrak{p}\infty_1$, where $\infty_1$ is one of the real places of $K$.