有限射影空间、线的几何展开与多量子比特
Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits
摘要：
给定GF（2）上的一个（2N - 1）维射影空间PG（2N - 1，2）及其直线的几何展布，存在这个空间到PG（N - 1，4）上的一个显著映射，其中展布的直线对应于PG（N-1，4）的直线对所张成的点和子空间。在这样的映射下，前一个空间的非退化二次曲面在后一个空间中有一个非奇异的厄米簇，这个二次曲面是{ \it双曲}或{ \it椭圆}，分别取决于N是{ \it偶数}或{ \it奇数}。我们利用这一性质表明，广义泡利群的N-量子位也形成两个不同的家庭根据奇偶性的N，并把对称运营商的作用到一个新的角度。取N=4的情况来说明该问题。
英文摘要：
Given a (2N - 1)-dimensional projective space over GF(2), PG(2N - 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N - 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N - 1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being {\it hyperbolic} or {\it elliptic} in dependence on N being {\it even} or {\it odd}, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric operators into a new perspective. The N=4 case is taken to illustrate the issue.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
