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      Dynamics of Entanglement Entropy from Einstein Equation

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          Abstract

          We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using the AdS/CFT. This is aimed at a first step to find a counterpart of Einstein equation in the CFT language. In particular, we point out that the entanglement entropy satisfies differential equations which directly correspond to the Einstein equation in several setups of AdS/CFT. We also define a quantity called entanglement density in higher dimensional field theories and study its dynamical property for weakly excited states in conformal field theories.

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          Area laws for the entanglement entropy - a review

          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.
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            Holographic Derivation of Entanglement Entropy from AdS/CFT

            , (2010)
            A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from AdS/CFT correspondence. We argue that the entanglement entropy in d+1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS_{d+2}, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal perfectly reproduces the correct entanglement entropy in 2D CFT when applied to AdS_3. We also compare the entropy computed in AdS_5 \times S^5 with that of the free N=4 super Yang-Mills.
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              Entanglement entropy and conformal field theory

              We review the conformal field theory approach to entanglement entropy. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries, and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.
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                Author and article information

                Journal
                26 April 2013
                2013-05-17
                Article
                10.1103/PhysRevD.88.026012
                1304.7100
                270251f4-19e3-4456-ae8d-9fa8fbf55310

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                YITP-13-29, IPMU-13-0083
                v2: 24 pages, calculations of translationally invariant limit have been corrected in section 4 and 5. References added
                hep-th cond-mat.str-el quant-ph

                Condensed matter,Quantum physics & Field theory,High energy & Particle physics

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