SIC-POVMs and Number Theory

Hilbert’s Twelfth Problem is a gargantuan programme being explored on dozens of research fronts in number theory.  The idea is to find easily-described algebraic numbers which generate the “abelian extensions” of a given number field F, just as roots of unity do in the case of such extensions of the rational numbers.  On the other hand the existence and construction of SIC-POVMs in general complex Hilbert space is a fascinating problem in quantum information theory.  Recent research has revealed natural generators of certain types of abelian fields occurring as structure constants in SICs, opening an exciting new quest for some universal object governing both SICs and these mysterious fields.