多粒子纠缠分类中的张量不变量
Tensor invariants for multipartite entanglement classification
AI总结：
本文探讨了多粒子纠缠结构的空间组织，通过trace-invariants与着色图的对应关系，提出了一种分类多粒子纯态LU轨道的方法，并分析了局部操作对纠缠资源理论的影响。
摘要：
组织多粒子量子系统的纠缠结构比双粒子情况更具挑战性：虽然双粒子纯态的局部酉（LU）轨道可通过其纠缠谱方便地表征，但多粒子纠缠结构的不变量相对难以定义和使用。根本原因在于双粒子问题可归结为矩阵不变量的分析，而多粒子问题则由更丰富的张量不变量空间所支配。本文通过所谓的trace-invariants探讨后者，这些不变量与已知的着色图有一一对应关系。我们首先解释了trace-invariant评估如何作为多粒子纯态LU轨道的标签，如何扩展到随机态，以及如何通过此类数据分析局部操作（LO）的影响。随后，我们关注参考态的一个（无限维）子空间内的纠缠分类，其基本构建块是不同维度的GHZ态。我们展示相对简单的trace-invariant子类足以区分参考态的LU轨道，并在LO（或LOCC）纠缠资源理论中实现完全（或不完全）表征。最后，我们研究如何在渐近大N展开中高效地区分参考态的（仍无限）子类，其中N是局部维度。这一分析依赖于与着色图相关的组合量，其中一些已在最近的随机张量文献中发挥了关键作用。在过程中报告了具有更广泛相关性的结果。
英文摘要：
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
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