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## Abstract

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.

Original language | English |
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Article number | 045003 |

Number of pages | 65 |

Journal | Reviews of Modern Physics |

Volume | 93 |

Issue number | 4 |

DOIs | |

Publication status | Published - 17 Dec 2021 |

## Austrian Fields of Science 2012

- 103025 Quantum mechanics
- 103012 High energy physics
- 103029 Statistical physics

## Projects

- 1 Active