SIC-POVMs from Stark units: Prime dimensions n2 + 3

Abstract

We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SIC-POVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SIC-POVM terminology) attached to such dimensions has fundamental units uK of norm −1. Let ZK denote the ring of integers of K; then, pZK splits into two ideals: p and p′. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields K(uK). Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ 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 ZK with modulus p∞1, where ∞1 is one of the real places of K.