SIC-POVMs and orders of real quadratic fields

Abstract

This paper concerns SIC-POVMs and their relationship to class field theory. SIC-POVMs are generalized quantum measurements (POVMs) described by $d^2$ equiangular complex lines through the origin in ${\mathbb{C}}^d$. Weyl--Heisenberg SICs are those SIC-POVMs described by the orbit a single vector under a finite Weyl--Heisenberg group $\mathrm{WH}(d)$. We relate known data on the structure and classification of Weyl--Heisenberg SICs in low dimensions to arithmetic data attached to certain orders of real quadratic fields. For $4\le d\le 90$, we show the number of known geometric equivalence classes of Weyl--Heisenberg SICs in dimension $d$ equals the cardinality of the ideal class monoid of the real quadratic order ${\mathcal{O}}_{{\Delta }_d}$ of discriminant ${\Delta }_d=(d+1)(d-3)$; we conjecture the equality extends to all $d\ge 4$. We prove that this conjecture implies the existence of more than one geometric equivalence class of Weyl--Heisenberg SICs for $d>22$. We conjecture Galois multiplets of SICs are in one-to-one correspondence with the over-orders ${\mathcal{O}}^{\prime }$ of ${\mathcal{O}}_{{\Delta }_d}$ in such a way that the number of classes in the multiplet equals the ring class number of ${\mathcal{O}}^{\prime }$. We test that conjecture against known data on exact SICs in low dimensions. We refine the class field hypothesis of Appleby, Flammia, McConnell, and Yard (arXiv:1604.06098) to predict the exact class field over $\mathbb{Q}(\sqrt{{\Delta }_d})$ generated by the ratios of vector entries for the equiangular lines defining a Weyl--Heisenberg SIC. The refined conjectures use a recently developed class field theory for orders of number fields (arXiv:2212.09177). The refined class fields assigned to over-orders ${\mathcal{O}}^{\prime }$ have a natural partial order under inclusion; the inclusions of these fields fail to be strict in some cases. We characterize such cases and give a table of them for $d<500$.

40 pages, 4 tables, 2 figures