Some comments on projective quadrics subordinate to pseudo--Hermitian spaces

关于集成 2×2 伪厄米矩阵与 PT 对称矩阵的分类和几何刻画

On Classification and Geometric Characterizations of Ensembled $2\times2$ Pseudo Hermitian and PT-Symmetric Matrices

本文对满足 H†G=GH 的 2×2 伪厄米矩阵进行分类，证明其均为 PT 对称矩阵，并利用线性簇、七个系综和四个单元给出几何刻画，揭示迹为零矩阵系综的行列式为二次曲面。

Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits

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 g…

On Invariant Notions of Segre Varieties in Binary Projective Spaces

Invariant notions of a class of Segre varieties $\Segrem(2)$ of PG(2^m - 1, 2) that are direct products of $m$ copies of PG(1, 2), $m$ being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains $\Segrem(2)$ and is invariant under its projective stabiliser group $\Stab{m}{2}$. By embedding PG(2^m - 1, 2) into \PG(2^m - 1, 4), a basis of the latter space is constructed that is invariant under $\Stab{m}{2}$ as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according…