Area laws for the entanglement entropy - a review

By Jens Eisert, M. Cramer, Martin Bodo Plenio. 2008

1,543 citations recorded by the source corpus.

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 we review the current status of area laws in these fields. 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, and disordered systems, non-equilibrium situations, classical correlation concepts, and topological entanglement entropies are discussed. A significant proportion of the article is devoted to the quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. We discuss matrix-product states, higher-dimensional analogues, and states from entanglement renormalization and conclude by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/628c980067574b0c4691a04c9b3ca5050a7bf970

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Pasquale Calabrese
2. T. Takayanagi
3. R. Myers
4. T. Nishioka
5. J. Cardy
6. Frank Verstraete
7. M. Fagotti
8. V. Alba

In Condensed Matter Physics

1. Pasquale Calabrese
2. J. Cardy
3. T. Takayanagi
4. T. Nishioka
5. Frank Verstraete
6. M. Fagotti
7. V. Alba
8. R. Myers

Citations in the corpus

• Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, background): “…(1) quantifies the amount of quantum correlations between the two regions, and as argued in the last section, this quantity is expected to be proportional to the surface of the boundary between the two regions, ∂A, and hence called the area law for the entanglement entropy (Eisert et al., 2010).”
• Vibrations, quanta and biology (2013, background): “t-DMRG will provide an accurate description for long times [152].”
• The Physics of Communicability in Complex Networks (2011, methodology): “…theory, where it has been used, for example, to establish area laws for the entanglement entropy; see, e.g., (Cramer and Eisert, 2006; Cramer et al., 2006; Hastings, 2004; Hastings and Koma, 2006; Schuch, 2007; Schuch et al., 2006) and especially (Eisert et al., 2010) for a comprehensive survey.”
• Quantum coherence and geometric quantum discord (2017, background): “(488) Let us point out that the “constant” term corresponds to the area law (Eisert et al., 2010; Hamma et al., 2005).”
• Space from Hilbert Space: Recovering Geometry from Bulk Entanglement (2016, background): “If we restrict ourselves to work at scales for which S ∝ A, all information encoded in the form of larger-scale entanglement is highly redundant.”