SIC-POVMs 和实二次域的阶数
SIC-POVMs and orders of real quadratic fields
AI总结：
本文研究SIC-POVMs与实二次域的算术数据之间的关系，发现低维Weyl-Heisenberg SICs的几何等价类数目与实二次域的理想类单oid的基数相等，并推测该关系可推广至所有d≥4。
摘要：
本文探讨SIC-POVMs及其与类域理论的关系。SIC-POVMs是通过原点的d²个等角复线描述的广义量子测量（POVMs）。Weyl-Heisenberg SICs是通过有限的Weyl-Heisenberg群WH(d)作用下的单向量轨道描述的SIC-POVMs。本文将已知的低维Weyl-Heisenberg SICs的结构和分类数据与某些实二次域的阶数数据联系起来。对于4≤d≤90，我们证明已知的Weyl-Heisenberg SICs在维度d的几何等价类数目等于实二次域O_{Δ_d}的理想类单oid的基数；我们推测该等式可扩展至所有d≥4。我们证明该推测意味着对于d>22，存在超过一个几何等价类的Weyl-Heisenberg SICs。我们推测SICs的Galois多重集与O'的过阶数O'之间存在一一对应关系，其中多重集的数目等于O'的环类数。我们通过低维精确SICs的已知数据测试该推测。我们改进了Appleby等人（arXiv:1604.06098）的类域假设，以预测由等角线定义的Weyl-Heisenberg SIC的精确类域。改进的推测使用了最近发展的数域阶数的类域理论（arXiv:2212.09177）。为过阶数O'分配的改进类域具有自然的包含偏序；在某些情况下，这些域的包含不严格。我们确定了这些情况并为d<500给出了表格。
英文摘要：
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 $\mathcal{O}_{Δ_d}$ of discriminant $Δ_d=(d+1)(d-3)$; 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 $\mathcal{O}_{Δ_d}$ 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 $\mathbb{Q}(\sqrt{Δ_d})$ 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 $d < 500$.
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