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Condensed Matter Applications of Entanglement Theory

Norbert Schuch

These are lecture notes from the 44th IFF Spring School "Quantum Information Processing" in Juelich, discussing applications of entanglement theory in condensed matter. The focus of the notes is on tensor network states, in particular Matrix Product States (MPS) and Projected Entangled Pair States (PEPS), which provide an efficient description of many-body states based on their entanglement structure; both numerical and analytical aspects of tensor networks are being covered. The notes close with a brief introduction to quantum complexity, which allows to assess the limitations to our ability to simulate quantum many-body systems.These are lecture notes from the 44th IFF Spring School "Quantum Information Processing" in Juelich, discussing applications of entanglement theory in condensed matter. The focus of the notes is on tensor network states, in particular Matrix Product States (MPS) and Projected Entangled Pair States (PEPS), which provide an efficient description of many-body states based on their entanglement structure; both numerical and analytical aspects of tensor networks are being covered. The notes close with a brief introduction to quantum complexity, which allows to assess the limitations to our ability to simulate quantum many-body systems.

arXiv:1306.5551Jun 24, 201321 citations

Les Houches Lecture Notes on Tensor Networks

Bram Vancraeynest-De Cuiper, Weronika Wiesiolek, Frank Verstraete

Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying entanglement structure and its associated (generalised) symmetries. In essence, tensor networks provide a compressed, holographic description of the complicated vacuum fluctuations in strongly correlated systems, and as such they break down the infamous many-body exponential wall. These lecture notes provide a concise overview of the most important conceptual, computational and mathematical aspects of this theory.Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying entanglement structure and its associated (generalised) symmetries. In essence, tensor networks provide a compressed, holographic description of the complicated vacuum fluctuations in strongly correlated systems, and as such they break down the infamous many-body exponential wall. These lecture notes provide a concise overview of the most important conceptual, computational and mathematical aspects of this theory.

cond-mat.str-elCross-listedarXiv:2512.24390SciPost Physics Lecture NotesDec 31, 20252 citations

Entanglement in many-body quantum systems

J. I. Cirac

Short review on entanglement, as seen from a quantum information perspective, and some simple applications to many-body quantum systems. Special emphasis in area laws, cold atoms, and efficient descriptions using tensor network states.

arXiv:1205.3742May 16, 20121 citation

Introduction to entanglement and applications to the simulation of many-body quantum systems

Mafalda L. Almeida, Yasser Omar, Vitor Rocha Vieira

WORLD SCIENTIFIC eBooksJul 1, 20071 citation

Hand-waving and interpretive dance: an introductory course on tensor networks

Jacob C. Bridgeman, Christopher T. Chubb

The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes. These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states. The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes. These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states. The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.

arXiv:1603.03039Journal of Physics A Mathematical and TheoreticalMay 10, 2017413 citations

Studying strongly correlated systems with tensor network state methods and quantum information theory

Mihály Máté

Jan 1, 20250 citations

Entanglement and tensor network states

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Entanglement and tensor network states

Abstract

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