Various problems concerning the geometry of the space $u^*(\cH)$ of Hermitian operators on a Hilbert space $\cH$ are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space $\cD(\cH)$ of density states on an $n$-dimensional Hilbert space $\cH$ is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space $\cD^k(\cH)$ of rank-$k$ states, $k=1,...,n$, is a smooth manifold of (real) dimension $2nk-k^2-1$ and this stratification is maximal in the sense that every smooth curve in $\cD(\cH)$, viewed as a subset of the dual $u^*(\cH)$ to the Lie algebra of the unitary group $U(\cH)$, at every point must be tangent to the strata $\cD^k(\cH)$ it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition $\cH=\cH^1\ot\cH^2$, an abstract criterion of entanglement is proved.
Geometry of entanglement witnesses for two qutrits
我们刻画了两量子态纠缠见证的凸子集。等价地，我们给出了3 x 3复矩阵的矩阵代数中正映射集的一个刻画。事实证明，这个集合的边界在SO（2）旋转方面表现出优雅的表示。我们推测，由旋转参数化的地图是最佳的，也就是说，他们提供了最强的工具来检测量子纠缠。作为一个副产品，我们发现了一类新的可分解的纠缠证人参数不适当的旋转正交群O（2）。
Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world)
我们提出了一种新的方法，四量子比特纠缠类的几何依赖于参数。更准确地说，我们使用不变理论和代数几何描述的各种层次的希尔伯特空间的SLOCC不变代数簇。Verstraete { \em et al.}被解释为可分离态集合的对偶簇的分量的稠密子集，并提出了一种基于四量子位量子态的不变量/协变量的算法来识别具有SLOCC等价范式（直到量子位置换）的状态。
