Area laws for the entanglement entropy - a review

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.

• The paper shows that entanglement entropy in quantum systems scales with the region's boundary rather than its volume, detailing rigorous results in both one-dimensional and higher-dimensional models.
• It employs advanced techniques such as Lieb-Robinson bounds and covariance matrices to derive analytical upper bounds for entanglement in various lattice configurations.
• The review connects area laws to tensor network methods and quantum simulations, highlighting implications for accurately modeling quantum many-body systems.

Overview of Area Laws for the Entanglement Entropy

The paper "Area laws for the entanglement entropy -- a review" by Eisert, Cramer, and Plenio explores the scaling behavior of entanglement entropy in quantum many-body systems. The concept of entanglement entropy, which measures the quantum correlations between a region and its complement in a quantum state, reveals insights into how these correlations decay and scale in different physical settings.

Core Ideas

The foundational aspect of the paper is the area law, which stipulates that the entanglement entropy of a subregion of a quantum many-body system often scales with the boundary of the region, rather than its volume. This is counterintuitive since one might expect an entropy that scales extensively with size, i.e., with volume, especially in thermal systems. The presence of area laws in various systems is meticulously reviewed, illustrating their ubiquity in both ground states of local Hamiltonians and in other unrelated fields such as black hole thermodynamics and holography.

Key Results

Several rigorous results for one-dimensional and higher-dimensional lattice models are discussed. For 1D systems:

• Gapped Systems: Ground states of gapped one-dimensional chains adhere to an area law. The lack of long-range quantum correlations due to a spectral gap implies that entanglement is localized around the boundaries of regions. This was proven in generality using techniques such as the Lieb-Robinson bounds.
• Critical Systems: Contrary to gapped systems, critical systems exhibit a logarithmic correction to the area law due to the presence of long-range correlations, as suggested by conformal field theory. The paper meticulously evaluates these using the concept of the Fermi surface in fermionic models or critical phenomena in conformal field theories.

For higher-dimensional systems, especially in two or three dimensions:

• Bosonic and Fermionic Models: While gapped models generally adhere to area laws, critical fermionic models show a potential violation characterized by logarithmic enhancements rather than strict area laws.
• Numerical and Analytical Bounds: Rigorous analytical expressions are presented for specific classes of Gaussian states, utilizing symplectic geometry and covariance matrices to provide upper bounds for entanglement entropy in terms of two-point correlation decay.

Implications and Observations

The study thoroughly discusses the implications of area laws for computational physics. The general adherence of quantum ground states to area laws or minor deviations (logarithmic corrections in critical systems) is directly related to the efficacy of tensor network methods such as matrix-product states (MPS) and the density-matrix renormalization group (DMRG). These methods capitalize on the limited entanglement to efficiently approximate ground states of local Hamiltonians.

Interestingly, area laws have parallels in classical physics, which the authors illuminate via mutual information in classical harmonic systems, showing similar boundary scaling predictions grounded in the locality of interactions.

Speculations and Further Directions

The paper speculates that understanding and characterizing systems that adhere or deviate from area laws can shed light on fundamental principles of quantum statistical mechanics and quantum field theory. Additionally, it points toward open problems in scaling behavior in the face of long-range interactions, disordered systems, and during non-equilibrium dynamics.

Conclusion

The extensive review shows that while area laws are a defining feature of many quantum systems, exceptions and variances in critical systems present a rich landscape for ongoing research. The connection between entanglement and topological phases, the effective degrees of freedom in simulations, and the conceptual linkage to black hole entropy provide a multifaceted interest in the study of entanglement entropy scaling.

This meticulous collation of results and discussion stands to inform further development of algorithms and theories within quantum many-body physics, quantum information, and related helioscopic ideas, continuing to bridge profound physical insights with computational feasibility in complex quantum systems.