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首页> 外文期刊>Journal of physics, A. Mathematical and theoretical >A sequence of qubit-qudit Pauli groups as a nested structure of doilies
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A sequence of qubit-qudit Pauli groups as a nested structure of doilies

机译：一组Qubit-Qudit Pauli群作为Doilies的嵌套结构

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Following the spirit of a recent work of one of the authors (2011 J. Phys. A: Math. Theor. 44 045301), the essential structure of the generalized Pauli group of a qubit-qudit, where d = 2k and an integer k ≥ 2, is recast in the language of a finite geometry. A point of such geometry is represented by the maximum set of mutually commuting elements of the group and two distinct points are regarded as collinear if the corresponding sets have exactly 2k - 1 elements in common. The geometry comprises 2 k - 1 copies of the generalized quadrangle of order 2 ('the doily') that form 2k - 1 - 1 pencils arranged into a remarkable nested configuration. This nested structure reflects the fact that maximum sets of mutually commuting elements are of two different kinds (ordinary and exceptional) and exhibits an intriguing alternating pattern: the subgeometry of the exceptional points of the (k + 2) case is found to be isomorphic to the full geometry of the k-case. It should be stressed, however, that these generic properties of the qubit-qudit geometry were inferred from purely computer-handled cases of k = 2, 3, 4 and 5 only and, therefore, their rigorous, computer-free proof for k ≥ 6 still remains a mathematical challenge.
机译：遵循其中一位作者的最新工作精神（2011 J. Phys。A：Math。Theor。44 045301），得出了qubit-qudit的广义Pauli群的基本结构，其中d = 2k，整数k ≥2，将以有限几何图形的语言重铸。这种几何形状的一个点由该组相互换向的元素的最大集合表示，并且如果相应的集合具有完全相同的2k-1个元素，则将两个不同的点视为共线。几何图形包含2 k-1阶2阶广义四边形的副本（“桌布”），形成2 k-1-1支铅笔，排列成显着的嵌套配置。这种嵌套结构反映了以下事实：互通元素的最大集合是两种不同的类型（普通和例外），并且呈现出一种有趣的交替模式：发现（k + 2）格的例外点的子几何与k盒的完整几何形状。但是，应该强调的是，仅从计算机处理的k = 2、3、4和5的情况下推断出qubit-qudit几何的这些一般属性，因此，它们对于k≥的严格的，无需计算机的证明6仍然是数学难题。

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