<i>Colloquium</i>: Area laws for the entanglement entropy

TLDR

In quantum many‑body systems, local interactions cause entanglement entropy to scale with the boundary area of a subregion rather than its volume, a phenomenon known as an area law. This Colloquium reviews the current status of area laws across black‑hole physics, quantum information, and many‑body physics. The review focuses on rigorous lattice‑model results in one and higher dimensions, contrasting bosonic and fermionic cases, and exploring connections to information‑propagation speed, disorder, non‑equilibrium, classical correlations, and topological entanglement. The authors highlight that area laws enable efficient numerical simulation via matrix‑product states and their higher‑dimensional analogues, and that entanglement renormalization schemes exploit these laws to reduce effective degrees of freedom.

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 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.