# nLab holographic entanglement entropy

Contents

## Surveys, textbooks and lecture notes

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

Holographic entanglement entropy refers to the expression of entanglement entropy of quantum field theories expressed “holographically” via a version of AdS-CFT duality in terms of the geometry of a higher-dimensional bulk spacetime

### Ryu-Takayanagi formula

For quantum field theories that are exhibited as boundary field theories on the asymptotic boundary $A$ of an approximately anti de Sitter spacetime via some approximation to AdS-CFT duality (for instance for QCD via AdS-QCD duality) their entanglement entropy of a given bounded domain $B\subset A$ turns out to be proportional to the area of the minimal-area surface inside the bulk spacetime that has the same boundary $\partial B$ (see Nishioka-Ryu-Takayanagi 09 (3.3) for review of the formula and Lewkowycz-Maldacena 13 for a conceptual explanation).

graphics grabbed from Nishioka-Ryu-Takayanagi 09

This relation is known as the Ryu-Takayanagi formula (Ryu-Takayanagi 06a, Ryu-Takayanagi 06b) for holographic computation of entanglement entropy, or holographic entanglement entropy, for short.

This is a generalization of the proportionality of black hole entropy to the area of its event horizon. Indeed, AdS-CFT duality applies to the near horizon geometry of black branes, the higher-dimensional generalizations of black holes and reduces 4d black holes under suitable KK-compactification (see also at black holes in string theory)

graphics grabbed from Nishioka-Ryu-Takayanagi 09

In fact quantum corrections to the black hole entropy in the presence of matter fields is equal to the entanglement entropy. (Ryu-Takayanagi 06a, p. 13)

Various properties of entanglement entropy find immediate geometric interpretations this way, for instance subadditivity

graphics grabbed from Nishioka-Ryu-Takayanagi 09

### Emergence of bulk spacetime from boundary information theory

Further discussion of implications of the Ryu-Takayanagi formula in van Raamsdonk 10 suggested that the logic may also be turned around: Instead of computing entanglement entropy of a given boundary field theory from known bulk geometry, conversely the bulk spacetime may be reconstructed from knowledge of the entanglement entropy of a boundary field theory.

Talking this perspective to the extreme suggests a description of bulk spacetimes entirely in terms of information theory/entanglement-relations of a boundary QFT (“tensor networks”, Swingle 09, Swingle 12, and quantum error correction codes ADH 14, PYHP 15, see Harlow 18 for review).

graphics grabbed from Harlow 18

graphics grabbed from Harlow 18

In this context the Ryu-Takayanagi formula for holographic entanglement entropy (above) has an exact proof PYHP 15, Theorem 2.

## References

### General

The original articles are

A proposal for a conceptual explanation is made in

Review:

Survey talks include

• Robert Myers, Holographic entanglement entropy, (pdf slides)

• Shinsei Ryu, Holographic geometry in Entanglement Renormalization (pdf slides)

• Juan Jottar, (Entanglement) Entropy in three-dimensional higher spin theories (pdf slides)

• Matthew Headrick, Entanglement entropies in holographic field theory (pdf slides)

• Tadashi Takayanagi, Entanglement Entropy and Holography (Introductory review) (pdf slides)

• Tom Hartmann, Entanglement entropy and geometry, talk slides, 2014 (pdf)

An influential argument that this relation implies that entanglement in the boundary theory is what makes spacetime as such appear in the bulk theory is due to

reviewed in

• Mark Van Raamsdonk, Lectures on Gravity and Entanglement, chapter 5 in New Frontiers in Fields and Strings

TASI 2015 Proceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics 2015 Theoretical Advanced Study Institute in Elementary Particle Physics (arXiv:1609.00026)

Discussion of the corresponding continuum theory, formulated via local nets of observables in algebraic quantum field theory:

Relation to renormalization of entanglement and tensor networks is due to

and further in terms of quantum error correcting codes due to

reviewed in

also

• Alexander Jahn, Marek Gluza, Fernando Pastawski, Jens Eisert, Majorana dimers and holographic quantum error-correcting codes, Phys. Rev. Research 1, 033079 (2019) (arXiv:1905.03268)

• Ahmed Almheiri, Holographic Quantum Error Correction and the Projected Black Hole Interior (arXiv:1810.02055)

• Alexander Jahn, Jens Eisert, Holographic tensor network models and quantum error correction: A topical review (arXiv:2102.02619)

• Felix M. Haehl, Eric Mintun, Jason Pollack, Antony J. Speranza, Mark Van Raamsdonk, Nonlocal multi-trace sources and bulk entanglement in holographic conformal field theories, J. High Energ. Phys. (2019) 2019: 005 (arxiv:1904.01584, talk recording)

• Han Yan, Hyperbolic Fracton Model, Subsystem Symmetry, and Holography, Phys. Rev. B 99, 155126 (2019) (arxiv:1807.05942)

Further development of these tensor networks in

Computation of black hole entropy in 4d via AdS4-CFT3 duality from holographic entanglement entropy in the ABJM theory for the M2-brane is discussed in

• Jun Nian, Xinyu Zhang, Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole (arXiv:1705.01896)

Discussion in terms of DHR superselection theory:

• Horacio Casini, Marina Huerta, Javier M. Magan, Diego Pontello, Entanglement entropy and superselection sectors I. Global symmetries (arXiv:1905.10487)

### Wilson lines computing holographic entropy in $AdS_3/CFT_2$

Discussion of BTZ black hole entropy and more generally of holographic entanglement entropy in 3d quantum gravity/AdS3/CFT2 via Wilson line observables in Chern-Simons theory:

• Martin Ammon, Alejandra Castro, Nabil Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP 10 (2013) 110 (arXiv:1306.4338)

• Jan de Boer, Juan I. Jottar, Entanglement Entropy and Higher Spin Holography in $AdS_3$, JHEP 1404:089, 2014 (arXiv:1306.4347)

• Alejandra Castro, Stephane Detournay, Nabil Iqbal, Eric Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 (arXiv:1405.2792)

• Mert Besken, Ashwin Hegde, Eliot Hijano, Per Kraus, Holographic conformal blocks from interacting Wilson lines, JHEP 08 (2016) 099 (arXiv:1603.07317)

• Andreas Blommaert, Thomas G. Mertens, Henri Verschelde, The Schwarzian Theory - A Wilson Line Perspective, JHEP 1812 (2018) 022 (arXiv:1806.07765)

• Ashwin Dushyantha Hegde, Role of Wilson Lines in 3D Quantum Gravity, 2019 (spire:1763572)

• Xing Huang, Chen-Te Ma, Hongfei Shu, Quantum Correction of the Wilson Line and Entanglement Entropy in the $AdS_3$ Chern-Simons Gravity Theory (arXiv:1911.03841)

• Eric D'Hoker, Per Kraus, Gravitational Wilson lines in $AdS_3$ (arXiv:1912.02750)

• Marc Henneaux, Wout Merbis, Arash Ranjbar, Asymptotic dynamics of $AdS_3$ gravity with two asymptotic regions (arXiv:1912.09465)

and similarly for 3d flat-space holography:

Discussion for 3d de Sitter spacetime:

• Alejandra Castro, Philippe Sabella-Garnier, Claire Zukowski, Gravitational Wilson Lines in 3D de Sitter (arXiv:2001.09998)