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Colloquium : Area laws for the entanglement entropy
Open accessAbstract
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.
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“Indeed, this fact is related to correlation and entanglement spreading according to the Lieb-Robinson theorem that states that, after a sufficiently large time, states can become strongly entangled ([91-96], see also [97]).”
From Can one trust quantum simulators? (2012) · cites this paper as [95]
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“I (n) A:B does not have all the correct properties to be an entanglement measure (see the review by Amico and Fazio in this special issue [8] for a discussion of various entanglement measures), but it has the fundamental property of satisfying the area law [6, 9], even at finite temperature [10].”
BackgroundThe classical-quantum boundary for correlations: Discord and related measures
Modi, K., Brodutch, A., Cable, H., Paterek, T., & Vedral, V. (2012) · Reviews of Modern Physics · doi:10.1103/revmodphys.84.1655 · Cited by 1,600
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Albash, T., & Lidar, D. A. (2018) · Reviews of Modern Physics · doi:10.1103/revmodphys.90.015002 · Cited by 1,546
Light-cone-like spreading of correlations in a quantum many-body system
Cheneau, M., Barmettler, P., Poletti, D., Endres, M., Schauß, P., Fukuhara, T., Gross, C., Bloch, I., Kollath, C., & Kuhr, S. (2012) · Nature · doi:10.1038/nature10748 · Cited by 864
Metadata
- Authors
- Eisert, J., Cramer, M., & Plenio, M. B.
- Affiliations
- University of Potsdam (DE)
- Venue
- Reviews of Modern Physics
- Publisher
- American Physical Society (APS)
- Published
- 2010-02-04
- Volume
- 82
- Issue
- 1
- Pages
- 277-306
- ISSN
- 0034-6861 (ISSN-L)
- 1539-0756
- OpenAlex
- W1922629554
- Cited by
- 2,873
- References
- 249
- Citations by year
- 2026: 164 · 2025: 232 · 2024: 229 · 2023: 204 · 2022: 225 · 2021: 196 · 2020: 206 · 2019: 205 · 2018: 169 · 2017: 163 · 2016: 166 · 2015: 158 · 2014: 163 · 2013: 141 · 2012: 99
- FWCI
- 64.56
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Reviews of Modern Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- publish.UP (University of Potsdam) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- area law, entanglement entropy, quantum many-body systems, lattice model, ground state, thermal states, disordered systems, non-equilibrium states, topological entanglement entropy, matrix product states, entanglement renormalization, correlation functions
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