Colloquium: Area laws for the entanglement entropy
Abstract
Physical interactions in quantum manybody 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 manybody 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. Matrixproduct states, higherdimensional 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.
 Publication:

Reviews of Modern Physics
 Pub Date:
 January 2010
 DOI:
 10.1103/RevModPhys.82.277
 arXiv:
 arXiv:0808.3773
 Bibcode:
 2010RvMP...82..277E
 Keywords:

 03.67.Mn;
 05.50.+q;
 05.70.a;
 Entanglement production characterization and manipulation;
 Lattice theory and statistics;
 Thermodynamics;
 Quantum Physics;
 Condensed Matter  Other Condensed Matter;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 28 pages, 2 figures, final version