The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many-body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. How matrix product states and projected entangled pair states describe many-body wave functions in terms of local tensors is reviewed. These tensors express how the entanglement is routed, act as a novel type of nonlocal order parameter, and the manner in which their symmetries are reflections of the global entanglement patterns in the full system is described. The manner in which tensor networks enable the construction of real-space renormalization group flows and fixed points is discussed, and the entanglement structure of states exhibiting topological quantum order is examined. Finally, a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications, is provided.

中文翻译：

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矩阵乘积状态和投影纠缠对状态：概念、对称性、定理

纠缠理论为描述多体系统中的相互作用和相关性提供了一种全新的语言。它的词汇表由量子位和纠缠对组成，语法由张量网络提供。回顾了矩阵乘积状态和投影纠缠对状态如何根据局部张量描述多体波函数。这些张量表达了纠缠的路由方式，作为一种新型的非局部有序参数，并且描述了它们的对称性是整个系统中全局纠缠模式的反映的方式。讨论了张量网络能够构建实空间重整化群流和不动点的方式，并检查表现出拓扑量子顺序的状态的纠缠结构。最后，