Proof of the holographic formula for entanglement entropy

Abstract

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.1 First indications on a possible relation between quantum entanglement and gravity phenomena should be attributed to authors of [3],[4] who suggested to interpret the Bekenstein-Hawking entropy as the entanglement entropy.

Entanglement entropy, trace anomalies and holography

“…The higher X (n) , just like the higher g (n) , are not universal, the reason being that their behavior under PBH transformations admits homogeneously transforming terms. 4 We succeeded therefore to put also this more general situation, with the action having two components, into a framework similar to the one we had for the bulk action alone. The action of the Weyl transformations on the boundary is embedded into bulk diffeomorphisms and world volume reparametrizations (3.19), (3.21).…”

“…where E (4) is the four-dimensional Euler density, C 2 (4) is the square of the four-dimensional Weyl tensor, G ij is the Einstein tensor (4.10)…”

“…The terms which could have δ-functions are the ones containing E (4) , G ij and C 2 (4) . There are no singular contributions from the φ field since the derivatives acting on it are not of a high enough order to produce a δ-function.…”

“…Arguments for the validity of this assumption were presented in [4]. Using the aforementioned holographic representation for massive d = 4 theories in [5] information about the phase structure of these theories was obtained; see also [6] for an earlier discussion of this issue.…”

Entanglement and nonunitary evolution

“…Further, a connection of the entanglement entropy to the evaluation of black hole entropy in string theory has been discussed in [26][27][28][29][30]. (See also [31][32][33][34][35][36][37] for a recent discussion of black hole entropy and entanglement in the context of AdS/CFT.) Thus, following [1], the current work is a natural next step in relating features of entanglement to black hole physics.…”