Projective ring line encompassing two-qubits
Projective ring line encompassing two-qubits
复制标题
包含两个量子位的投影环线
DOI：
10.1007/s11232-008-0076-x
复制
发表时间：
2006
影响因子：
1
通讯作者：
P. Pracna
中科院分区：
文献类型：
--
作者：
M. Saniga;M. Planat;P. Pracna
我们发现，在GF（2）中系数的2×2矩阵的（非交换）环上的投影线完全容纳了表征两量子比特系统的15个算子（广义泡利矩阵）的代数。相关的子配置由15个点组成，每个点都同时远离或同时邻近线上的（任何）两个给定的远点。算子可以与点一一对应，使得它们的对易关系可以由点的基本几何精确地再现，其中近邻和远距的环几何概念对应于相应的对易和非对易操作概念。这个显著的配置可以用两种主要不同的方式来看待，解释观测量代数的基本对应的9+6和10+5因子分解：首先，作为GF（2）× GF（2）上的投影线的不相交并。（“Mermin”部分）和GF（4）上经过省略的两个选定点的两条线;第二，作为二阶广义四边形，其卵形和/或扩展对应于五个相互非交换算子的（最大）集合和/或五个最大交换子集的三个算子的组。这些发现为有限维量子系统的代数几何建模开辟了意想不到的可能性，并为其众多应用开辟了全新的前景。
We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF(2) × GF(2) (the “Mermin” part) and two lines over GF(4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications.