Maclachlan-Martin Kleinian群的几何背景
Geometric contextuality from the Maclachlan-Martin Kleinian groups
摘要：
存在多个量子比特的上下文集合，其交换由于双生成器自由群$G=\left\langle x,y\right\rangle$的子群$H$的陪集几何结构$\mathcal{G}$而被参数化。本文证明了由两个椭圆等距$\mathcal{G}$和$g$（Martin-Maclachlan分类）生成的非紧算术Kleinian子群$K=\left\langle f,g\right\rangle$是合适的上下文过滤器.标准的上下文几何体，如一些薄的广义多边形（从Mermin的$3 \times 3$网格开始）属于此框架。定义在虚二次域$O\_d=\mathbb{Q}(\sqrt{-d})$上的Bianchi群$PSL(2,O\_d)$、$d \in \{1,3\}$起着特殊的作用。
英文摘要：
There are contextual sets of multiple qubits whose commutation is parametrized thanks to the coset geometry $\mathcal{G}$ of a subgroup $H$ of the two-generator free group $G=\left\langle x,y\right\rangle$. One defines geometric contextuality from the discrepancy between the commutativity of cosets on $\mathcal{G}$ and that of quantum observables.It is shown in this paper that Kleinian subgroups $K=\left\langle f,g\right\rangle$ that are non-compact, arithmetic, and generated by two elliptic isometries $f$ and $g$ (the Martin-Maclachlan classification), are appropriate contextuality filters. Standard contextual geometries such as some thin generalized polygons (starting with Mermin's $3 \times 3$ grid) belong to this frame. The Bianchi groups $PSL(2,O\_d)$, $d \in \{1,3\}$ defined over the imaginary quadratic field $O\_d=\mathbb{Q}(\sqrt{-d})$ play a special role.
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