Towards a Theory of SIC-like Phenomena: Regular Bouquets and Generalised Heisenberg Groups

Abstract

We lay the foundations for a broad algebraic theory encompassing SICs in the hope of elucidating their heuristic connections with Stark units. What emerges is a greatly generalised set-up with added structure and potential for applications in other areas. Let $A$ and $B$ be finite modules for a commutative ring $R$, $C$ a finite abelian group and $\lambda: A\times B\rightarrow C$ an $R$-balanced bilinear pairing. The main constructs are the generalised Heisenberg group ${\cal H}={\cal H}(A,B,C,\lambda)$ attached to these data (an abstract central extension of $A\oplus B$ by $C$) which plays the role of the Weyl-Heisenberg group in SIC theory, together with its canonical, unitary Schr\"odinger representations. The SIC itself is replaced by an ${\cal H}$-orbit of complex lines in the representation space, termed a `bouquet'. The overlaps of the SIC are interpreted as a map from ${\cal H}/Z({\cal H})$ into $\mathbb C$ whose absolute values are the Hermitian `angles' between the lines in the bouquet. We also introduce a regularity condition on bouquets in terms of the angle-map, intended to weaken the equiangularity condition of SICs. At the same time, it allows the incorporation of the $R$-structure via the abstract automorphism group of ${\cal H}$ which in turn generalises the Clifford group of SIC theory via its associated Weil representation. As well as several subsidiary definitions and `structural' results, we prove a new `clinometric relation' for the angle-map, determine the structure of the automorphism group and introduce a large class of examples of arithmetic origin, derived from the trace-pairing on quotients of fractional ideals in an arbitrary number field, which we investigate in greater detail.