A generalized algebraic framework for SIC-like configurations. From finite modules A and B over a commutative ring, a finite abelian group C, and an R-balanced bilinear pairing A x B -> C, one builds a generalized Heisenberg group (a central extension of A + B by C) that plays the role of the Weyl-Heisenberg group in SIC theory, together with its unitary Schrodinger representations. A SIC is replaced by a bouquet (an orbit of complex lines in the representation space), equiangularity is weakened to a regularity condition on the associated angle-map, and a large class of arithmetic examples arises from the trace pairing on quotients of fractional ideals in a number field.
This paper lays the algebraic foundations for a broad generalization of Symmetric Informationally Complete measurements (SICs) in quantum information, motivated by their conjectural links to Stark un…