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

• URL: http://arxiv.org/abs/2112.05552v2
• Date: Wed, 10 Aug 2022 11:38:06 GMT
• Title: SIC-POVMs from Stark units: Prime dimensions n^2+3
• Authors: Marcus Appleby, Ingemar Bengtsson, Markus Grassl, Michael Harrison, Gary McConnell
• Abstract summary: We show a recipe for constructing a SIC fiducial vector in complex Hilbert space of dimension of the form $d=n2+3$. Such structures are shown to exist in thirteen prime dimensions of this kind, the highest being $p=19603$.
• Score: 0.41998444721319206
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $K$.

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