单形的Moebius对与交换Pauli算子
Moebius Pairs of Simplices and Commuting Pauli Operators
摘要：
存在一大类作用于希尔伯特空间的算子群，其中群元素的交换性可以用嵌入射影空间PG（$n, p$）中的辛极空间的几何语言来表示，$n$是奇数，而$p$是素数。在这里，我们基于$n$ -单形的Moebius对的存在性，给出了关于交换和非交换群元的一个结果，即.例如，成对的$n$ -单形，它们彼此是\emph{mutually inscribed and circumscribed}。对于表示一个$n$ -单纯形的群元素，在中心之外没有元素可以与所有的群元素交换。这允许在群论术语中表达相关联的极空间的维度$n$。根据我们的构造，任何$n$ -单形的Moebius对对应于两个不相交的群元素（算子）族，具有以下性质：（i）同一族的任何两个不同的元素不交换。(ii)一个族中的每个图元与另一个族中除一个图元外的所有图元都可以互换。一个三量子比特的广义泡利群作为一个非平凡的例子来说明$p=2$和$n=5$的理论。
英文摘要：
There exists a large class of groups of operators acting on Hilbert spaces, where commutativity of group elements can be expressed in the geometric language of symplectic polar spaces embedded in the projective spaces PG($n, p$), $n$ being odd and $p$ a prime. Here, we present a result about commuting and non-commuting group elements based on the existence of so-called Moebius pairs of $n$-simplices, i. e., pairs of $n$-simplices which are \emph{mutually inscribed and circumscribed} to each other. For group elements representing an $n$-simplex there is no element outside the centre which commutes with all of them. This allows to express the dimension $n$ of the associated polar space in group theoretic terms. Any Moebius pair of $n$-simplices according to our construction corresponds to two disjoint families of group elements (operators) with the following properties: (i) Any two distinct elements of the same family do not commute. (ii) Each element of one family commutes with all but one of the elements from the other family. A three-qubit generalised Pauli group serves as a non-trivial example to illustrate the theory for $p=2$ and $n=5$.
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