SIC-POVMs from Stark Units

We propose a recipe for constructing a SIC fiducial vector in any complex Hilbert space whose dimension is of the form d = n + 3, starting from Stark units in a ray class field that does not contain a complex root of unity. The recipe relies on some number theoretical theorems, a version of the Stark conjectures, and some standard conjectures for SICs. In this paper we focus on the conceptually simplest case, prime dimensions of the form d = n+3, and report that we have constructed SICs in twelve prime dimensions of this kind, the highest being d = 19603.

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