SIC-POVMs and orders of real quadratic fields

TL;DR

This paper explores the deep connection between SIC-POVMs in quantum information theory and class field theory of real quadratic fields, proposing conjectures and refinements supported by computational data.

Contribution

It relates Weyl--Heisenberg SIC classifications to ideal class monoids of real quadratic orders and refines class field hypotheses with new theoretical insights.

Findings

Number of SIC classes matches ideal class monoid cardinality for 4 ≤ d ≤ 90.

Conjecture that this equality extends to all d ≥ 4.

Refined class field predictions for ratios of vector entries in SICs.

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 ${\rm WH}(d)$. 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 $\mathcal{O}'$ of $\mathcal{O}_{\Delta_d}$ in such a way that the number of classes in the multiplet equals the ring class number of $\mathcal{O}'$. 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 $\mathcal{O}'$ 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 $d < 500$.