arXiv 1503.060712015-03quant-phmath-phmath.MP

多体纠缠测度

Multipartite entanglement measures

Szilárd Szalay

备注：: 40 pages, 2 tables, 2 figures. v4: small fix in fig.2, does not affect results
发表信息：: Phys. Rev. A 92, 042329 (2015)
DOI：: 10.1103/PhysRevA.92.042329

摘要：

本文主要研究如何定义混合量子态的多体纠缠度。由于部分可分性和多体纠缠的结构变得复杂，如果子系统的数量超过两个，人们不能期望存在一个最终的标量纠缠度量，它掌握了即使是一小部分的丰富的多体纠缠的层次结构，和一些更高阶的结构特征是必要的。在本文中，我们朝着这个方向迈出了一些步伐。首先，我们揭示了部分可分性分类的格理论结构，介绍了较早的[Sz。Szalay和Z. Kokenyesi，Phys.Rev.A86，032341（2012）]。事实证明，从数学上讲，纠缠类的结构是不同类型的部分可分性结构的上集格，而后者是子系统分区格的下集格。事实证明，从物理上讲，这种结构与LOCC可转换性有关：如果一个类的状态可以映射到另一个类，那么这个类可以在层次结构中找到。其次，我们引入了多体单调性的概念，表示一组给定的纠缠单调性，同时测量不同种类的纠缠，也表现出相同的层次结构的纠缠类。然后，我们构建了这样的层次的纠缠度量，并提出了一个物理上很好的动机，是直接的多体广义的纠缠形成的基础上的纠缠熵，由统计可解释性的概念。这组测度所表现出的多体单调性促使我们考虑这些测度是某种“统一”纠缠概念的不同表现。

英文摘要：

The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one can not expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher order structure characterizing that is needed. In this paper we make some steps towards this direction. First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. Kokenyesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It turns also out that, physically, this structure is related to the LOCC convertibility: If a state from a class can be mapped into another one, then that class can be found higher in the hierarchy. Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures, and propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some "unified" notion of entanglement.

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