格拉斯曼星系的恒星表象
Stellar Representation of Grassmannians
摘要：
纯量子自旋- $s$态可以用球上的$2s$点来表示，如Majorana在1932年所示--这种描述在研究态的旋转对称性和许多其他性质时被证明特别有用，因为当态在希尔伯特空间中经历$SU(2)$变换时，这些点在球上刚性旋转。与此同时，Wilzcek-Zee效应，它涉及希尔伯特空间的退化$k$维线性子空间的循环演化，以及由Schroedinger方程决定的相关完整性，已经被提出作为实现逻辑门的容错机制，在量子计算中具有应用。我们表明，在本文中，如何描述这样的子空间的马约拉纳一样的点集的领域，也旋转刚性下$SU(2)$变换-建设实际上是有效的任意完全反对称$k$ -partite量子态。
英文摘要：
Pure quantum spin-$s$ states can be represented by $2s$ points on the sphere, as shown by Majorana in 1932 --- the description has proven particularly useful in the study of rotational symmetries of the states, and a host of other properties, as the points rotate rigidly on the sphere when the state undergoes an $SU(2)$ transformation in Hilbert space. At the same time, the Wilzcek-Zee effect, which involves the cyclic evolution of a degenerate $k$-dimensional linear subspace of the Hilbert space, and the associated holonomy dictated by Schroedinger's equation, have been proposed as a fault-tolerant mechanism for the implementation of logical gates, with applications in quantum computing. We show, in this paper, how to characterize such subspaces by Majorana-like sets of points on the sphere, that also rotate rigidly under $SU(2)$ transformations --- the construction is actually valid for arbitrary totally antisymmetric $k$-partite qudit states.
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