Proof of the holographic formula for entanglement entropy

By D. Fursaev. 2006

304 citations recorded by the source corpus.

Entanglement entropy for a spatial partition of a quantum system is studied in theories which admit a dual description in terms of the anti-de Sitter (AdS) gravity one dimension higher. A general proof of the holographic formula which relates the entropy to the area of a codimension 2 minimal hypersurface embedded in the bulk AdS space is given. The entanglement entropy is determined by a partition function which is defined as a path integral over Riemannian AdS geometries with non-trivial boundary conditions. The topology of the Riemannian spaces puts restrictions on the choice of the minimal hypersurface for a given boundary conditions. The entanglement entropy is also considered in Randall-Sundrum braneworld models where its asymptotic expansion is derived when the curvature radius of the brane is much larger than the AdS radius. Special attention is payed to the geometrical structure of anomalous terms in the entropy in four dimensions. Modification of the holographic formula by the higher curvature terms in the bulk is briefly discussed.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/f3a3538ba7569ee58bcccd76beb51301271fc7ce

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. T. Takayanagi
2. R. Myers
3. V. Hubeny
4. D. Harlow
5. M. Rangamani
6. T. Nishioka
7. T. Faulkner
8. M. Headrick

In Geometry and Topology

1. R. Myers
2. T. Takayanagi
3. V. Hubeny
4. M. Headrick
5. M. Rangamani
6. William Donnelly
7. T. Nishioka
8. H. Casini

In Nuclear and High Energy Physics

1. T. Takayanagi
2. R. Myers
3. V. Hubeny
4. T. Nishioka
5. M. Rangamani
6. D. Harlow
7. T. Faulkner
8. Xi Dong

Citations in the corpus

• Holographic entanglement entropy for general higher derivative gravity (2013, unknown): “scription was conjectured by Ryu and Takayanagi [5] and proven by Lewkowycz and Maldacena [7]. Proof of the formula for special cases includes [6,18,19]. Earlier attempts to prove the formula include [20,21]. One-loop corrections were analyzed in [8,9]. A covariant version of the prescription which ”
• Fractional quantum Hall effect via holography: Chern-Simons, edge states and hierarchy (2009, background): “The action principle tells us that γ is the minimal area surface [41, 40] whose boundary coincides with ∂A.”
• Entanglement entropy from a holographic viewpoint (2012, methodology): “See also [49, 107] for other aspects of higher derivative corrections to the holographic entanglement entropy.”
• Comments on holographic entanglement entropy and RG flows (2012, background): “where v ∼ V indicates that v is a bulk surface that is homologous to the boundary region V [80, 81].”
• Causal holographic information (2012, background): “The first attempt to derive the minimal surface proposal was made in [6], which was critically examined in [7], who pointed out some loopholes in the above derivation.”
• Comments on quantum gravity and entanglement (2009, background): “Thanks to Ryu and Takayanagi [26] (see also [27, 28, 29, 30]), we have an explicit proposal for the bulk interpretation of entanglement entropy: it is given by”
• AdS bubbles, entropy and closed string tachyons (2006, background): “Refer to [46] for its proof from the basic principle of the AdS/CFT correspondence.”
• Geometric entropy and edge modes of the electromagnetic field (2015, unknown): “The derivations of the RT formula [43, 51, 52] involve the geometric entropy (or the related replica trick [53]) and therefore one expects that RT calculates the geometric entropy.”
• Holographic mutual information is monogamous (2011, background): “) It has been conjectured [22, 4] that, in the presence of such corrections, the entanglement entropy is still given by a minimization over surfaces, but with the area functional itself corrected by higher-derivative terms:”
• Holographic calculations of Rényi entropy (2011, methodology): “However, attempting to work with this singular bulk space in a straightforward way [13] produces incorrect results [7].”
• Distributional Geometry of Squashed Cones (2013, methodology): “Subadditivity property of entanglement entropy and arguments of [14] require that Σ̃ is a hypersurface at which the functional in the r.”
• The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT (2013, background): “In principle one should be able to derive it using just these rules, however the attempt in [19] failed as was emphasized in [20].”
• Holographic c-theorems in arbitrary dimensions (2010, result): “As emphasized in [75], the derivation presented in [76] is flawed and, for example, leads to incorrect results in holographic calculations of Renyi entropies.”
• Entanglement Entropy of Black Holes (2011, background): “This proposal is very interesting since it allows one to compute the entanglement entropy in a purely geometrical way (see also [109]).”
• Holographic evolution of entanglement entropy (2010, background): “According to [41], the geodesic whose length should give the entanglement entropy of a certain region does not need to be connected.”