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Title: Linearity of holographic entanglement entropy

Abstract

Here, we consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of entropy operators in general systems with a large number of degrees of freedom.

Authors:
 [1];  [2];  [1]
  1. Stanford Univ., Stanford, CA (United States)
  2. Institute for Advanced Study, Princeton, NJ (United States)
Publication Date:
Research Org.:
Stanford Univ., CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science Foundation (NSF)
Contributing Org.:
Princeton Univ., NJ (United States)
OSTI Identifier:
1358372
Alternate Identifier(s):
OSTI ID: 1356081
Grant/Contract Number:
SC0009988; PHY-1316699
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 2; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; level-spacing distributions; black-holes; airy kernel; quantum; system; states; AdS-CFT correspondence; black holes in string theory; gauge-gravity correspondence; conformal field theory; AdS-CFT Correspondence

Citation Formats

Almheiri, Ahmed, Dong, Xi, and Swingle, Brian. Linearity of holographic entanglement entropy. United States: N. p., 2017. Web. doi:10.1007/JHEP02(2017)074.
Almheiri, Ahmed, Dong, Xi, & Swingle, Brian. Linearity of holographic entanglement entropy. United States. doi:10.1007/JHEP02(2017)074.
Almheiri, Ahmed, Dong, Xi, and Swingle, Brian. Tue . "Linearity of holographic entanglement entropy". United States. doi:10.1007/JHEP02(2017)074. https://www.osti.gov/servlets/purl/1358372.
@article{osti_1358372,
title = {Linearity of holographic entanglement entropy},
author = {Almheiri, Ahmed and Dong, Xi and Swingle, Brian},
abstractNote = {Here, we consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of entropy operators in general systems with a large number of degrees of freedom.},
doi = {10.1007/JHEP02(2017)074},
journal = {Journal of High Energy Physics (Online)},
number = 2,
volume = 2017,
place = {United States},
year = {Tue Feb 14 00:00:00 EST 2017},
month = {Tue Feb 14 00:00:00 EST 2017}
}

Journal Article:
Free Publicly Available Full Text
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Cited by: 4works
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  • Here, we consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certainmore » such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of entropy operators in general systems with a large number of degrees of freedom.« less
  • A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (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{sub d+2}, analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS{sub 3}. We also compare the entropy computed in AdS{sub 5}xS{sup 5} with that of the free N=4 super Yang-Mills theory.
  • We study the holographic representation of the entanglement entropy, recently proposed by Ryu and Takayanagi, in a braneworld context. The holographic entanglement entropy of a de Sitter brane embedded in an anti-de Sitter (AdS) spacetime is evaluated using geometric quantities, and it is compared with two kinds of de Sitter entropy: a quarter of the area of the cosmological horizon on the brane and entropy calculated from the Euclidean path integral. We show that the three entropies coincide with each other in a certain limit. Remarkably, the entropy obtained from the Euclidean path integral is in precise agreement with themore » holographic entanglement entropy in all dimensions. We also comment on the case of a five-dimensional braneworld model with the Gauss-Bonnet term in the bulk.« less
  • When a quantum system is divided into subsystems, their entanglement entropies are subject to an inequality known as strong subadditivity. For a field theory this inequality can be stated as follows: given any two regions of space A and B, S(A)+S(B){>=}S(A cup B)+S(A intersection B). Recently, a method has been found for computing entanglement entropies in any field theory for which there is a holographically dual gravity theory. We give a simple geometrical proof of strong subadditivity employing this holographic prescription.
  • Cited by 6