Colloquium: Area laws for the entanglement entropy

2008/08/31 by Jens Eisert, J. Eisert, M. Cramer +2 · 2,905 citations
Mathematics · Physics and Astronomy · #Entropy (arrow of time) #Law #Physics #Physics of Superconductivity and Magnetism #Quantum #Quantum and electron transport phenomena #Quantum discord #Quantum entanglement #Quantum many-body systems #Quantum mechanics #Statistical physics #Theoretical physics #cond-mat.other #hep-th #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1103/revmodphys.82.277

published in Reviews of Modern Physics 82(1), 277-306 (American Physical Society) · 28 pages, 2 figures, final version

openalex publication_date 2010/02/04 · arxiv created 2010/02/12 · arxiv updated 2011/01/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

Citations

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• Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence
• Fault-tolerant quantum computation by anyons
• Two soluble models of an antiferromagnetic chain
• The world as a hologram
• Quantum entanglement

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