Canonical forms of two-qubits under the action of stochastic local operations and classical communications (SLOCC) offer great insight for understanding nonlocality and entanglement shared by them. They also enable geometric picture of two-qubit states within the Bloch ball. It has been shown [Phys. Rev. A 64, 010101(R) (2001)] that an arbitrary two-qubit state gets transformed under SLOCC into one of the two different canonical forms. One of these happens to be the Bell diagonal form of two-qubit states and the other a nondiagonal canonical form is obtained for a family of rank deficient two-qubit states. The method employed by Verstraete et al. [Phys. Rev. A 64, 010101(R) (2001)] required highly nontrivial results on matrix decompositions in $n$-dimensional spaces with an indefinite metric. Here we employ an entirely different approach—inspired by the methods developed by Rao et al. [J. Mod. Opt. 45, 955 (1998)] in classical polarization optics—which leads naturally towards the identification of two inequivalent SLOCC invariant canonical forms for two-qubit states. In addition, our approach results in a simple geometric visualization of two-qubit states in terms of their SLOCC canonical forms.

中文翻译：

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本地操作下的两个量子位态的规范形式

在随机本地操作和经典通信（SLOCC）的作用下，两个量子位的规范形式为理解它们的非局部性和纠缠提供了很好的见识。它们还使能Bloch球中两个量子位状态的几何图形。它已经显示[ Phys。修订版A 64，010101（R）（2001）]，一个任意的双量子位状态被随机LOCC下转化成的一个2种不同规范化形式。其中之一恰好是两个量子位状态的贝尔对角线形式，而另一种是针对秩不足的两个量子位状态族获得的非对角正则形式。Verstraete等人采用的方法。[物理 版本A 64，010101（R）（2001）]要求在矩阵分解中获得非常重要的结果 $ñ$具有不确定度量的三维空间。在这里，我们采用了一种完全不同的方法-受Rao等人开发的方法的启发。[ J. Mod。选择。 45，955（1998）]在经典偏振光学-这导致自然朝向2不等价随机LOCC的识别不变量双量子位状态规范化形式。此外，我们的方法可以根据SLOCC规范形式对两个量子位状态进行简单的几何可视化。