تقرير
SIC-POVMs and orders of real quadratic fields
| العنوان: | SIC-POVMs and orders of real quadratic fields |
|---|---|
| المؤلفون: | Kopp, Gene S., Lagarias, Jeffrey C. |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics Quantum Physics |
| مصطلحات موضوعية: | Mathematics - Number Theory, Mathematics - Metric Geometry, Quantum Physics, 11R37 (Primary), 11R29, 11R65, 81P18, 81P15, 81R05, 42C15 (Secondary) |
| الوصف: | We consider the problem of counting and classifying symmetric informationally complete positive operator-valued measures (SICs or SIC-POVMs), that is, sets of $d^2$ equiangular lines in $\mathbb{C}^d$. For $4 \leq d \leq 90$, we show the number of known equivalence classes of Weyl--Heisenberg covariant SICs in dimension $d$ equals the cardinality of the ideal class monoid of (not necessarily invertible) ideal classes in the real quadratic order of discriminant $(d+1)(d-3)$. Equivalently, this is the number of $\mathbf{GL}_2(\mathbb{Z})$ conjugacy classes in $\mathbf{SL}_2(\mathbb{Z})$ of trace $d-1$. We conjecture the equality extends to all $d \geq 4$. We prove that this conjecture implies more that one equivalence class of Weyl--Heisenberg covariant SICs for every $d > 22$. Additionally, we refine the "class field hypothesis" of Appleby, Flammia, McConnell, and Yard (arXiv:1604.06098) to predict the exact class field generated by the ratios of vector entries for the equiangular lines defining a Weyl--Heisenberg covariant SIC. The class fields conjecturally associated to SICs in dimension $d$ come with a natural partial order under inclusion; we show that the natural inclusions of these fields are strict, except in a small family of cases. Comment: 45 pages, 3 tables |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.08048 |
| رقم الأكسشن: | edsarx.2407.08048 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |