Convexity of momentum map, Morse index, and quantum entanglement
Convexity of momentum map, Morse index, and quantum entanglement
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DOI：
10.1142/s0129055x14500044
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发表时间：
2012-08
影响因子：
1.8
通讯作者：
A. Sawicki;Michał Oszmaniec;M. Ku's
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文献类型：
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作者：
A. Sawicki;Michał Oszmaniec;M. Ku's
本文从拓扑学的角度分析了复合量子系统所有纯态SLOCC （Stochastic Local Operations with Classical Communication）类的空间。我们对可区分和不可区分的粒子都做了实验。一般来说，这个空间的拓扑结构比较复杂，因为它是一个非hausdorff空间。利用几何不变理论（GIT）和动量映射几何，我们提出了一种将所有SLOCC类的空间划分为数学上和物理上有意义的族的方法。每个族可能由许多“渐近”等价的SLOCC类组成。此外，每一个都包含一个独特的SLOCC类，在这个类上状态Var[v]的总方差（一种定义良好的纠缠度量）达到最大值。我们提供了一种寻找Var[v]的关键集的算法，该算法利用动量映射的凹凸性，并允许对这些定义的SLOCC类族进行分类。家庭的数量通常是无限的。我们使用被称为Kirwan-Ness分层的动量图几何中的一些发展，对有限多组族引入了额外的细化。我们还讨论了如何使用应用于SLOCC类的动量映射的凸性来等效地定义它。此外，我们注意到总状态方差临界集的莫尔斯指数可以解释纠缠增加的非slocc方向的数量，并对几个示例系统进行了计算。最后，我们引入了sloc不变纠缠测度作为临界点处状态总方差的平方根，并解释了其几何意义。
We analyze from the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry, we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many "asymptotically" equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well-defined measure of entanglement) of the state Var[v] attains maximum. We provide an algorithm for finding critical sets of Var[v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using some developments in the momentum map geometry known as the Kirwan–Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.