Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.19711 · Physical review. D/Physical review. D.

How Einstein’s equations emerge from CFT 2

Xin Jiang, Peng Wang, Houwen Wu, Haitang Yang

Published Oct 13, 2025• 1 citation
arXivHTMLPDF

Abstract

The of the entanglement entropies between disjoint subsystems enables us to show that the dynamical equation of the entanglement entropy in CFT 2 is precisely the three-dimensional Einstein equation. We establish a profound relation between the cosmological constant and CFT 2 entanglement entropy. Thus, entanglement entropies induce internal gravitational geometries in CFT 2 . Extracting the dual metric from an entanglement entropy becomes a straightforward procedure. Remarkably, we discover that the renormalization group equation is a geometric identity.

DOI
10.1103/zg5x-34mn

Related papers

Entanglement entropy, holography and gravity

L.R. de Ruiter

In this thesis we review how entanglement entropy is related to the gravitational equations of motion by means of the AdS/CFT correspondence. We consider the entanglement entropy of a ball-shaped spatial region in the CFT vacuum and derive a holographic relation between the relative entropy and the equations of motion in the dual bulk spacetime. For small perturbations around the CFT vacuum, we show that the first law of entanglement entropy requires the dual metric to satisfy the linearized Einstein equation when the entanglement entropy is given by the area of an extremal surface in the bulk. We discuss a recent paper in which the complete Einstein equation is derived and explain why we believe this result is incomplete. This thesis serves both as a commentary on this paper, as well as a detailed review of the relationship between entanglement and gravity.

•Jan 1, 2018•0 citations

Emergent Spacetime Geometry from Quantum Information Correlations

Laura Edreí Salazar Cuevas

We propose an operational framework in which spacetime geometry and gravitational dynamics emerge directly from quantum information correlations in many-body systems. Utilizing Multiscale Entanglement Renormalization Ansatz (MERA), we derive an effective metric based on mutual information, demonstrating that Einstein's field equations arise from a variational entropy principle, providing a theoretical link between entanglement density and the cosmological constant.

•Jan 9, 2026•0 citations

Timelike entanglement entropy in dS3/CFT2

Xin Jiang, Peng Wang, Houwen Wu +1 more

Abstract In the context of dS3/CFT2, we propose a timelike entanglement entropy defined by the renormalization group flow. This timelike entanglement entropy is calculated in CFT by using the Callan-Symanzik equation. We find an exact match between this entanglement entropy and the length of a timelike geodesic connecting two different spacelike surfaces in dS3. The counterpart of this entanglement entropy in AdS3 is a spacelike one, also induced by RG flow and extends all the way into the bulk of AdS3. As a result, in both AdS3/CFT2 and dS3/CFT2, there exist exactly three entanglement entropies, providing precisely sufficient information to reconstruct the three-dimensional bulk geometry.

arXiv:2304.10376•Journal of High Energy Physics•Aug 31, 2023•34 citations

Derive Einstein equation from CFT entanglement entropy

Xin Jiang, Haitang Yang

We explicitly show how to derive the $(D+1)$-dimensional Einstein equation from the entanglement entropy between codimension-one {\it disjoint} regions in $D$-dimensional conformal field theory.

arXiv:2510.27250•Oct 31, 2025•0 citations

Bulk emergence and the RG flow of entanglement entropy

Thomas Faulkner

We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional bulk modular hamiltonian in an emergent AdS space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein’s equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.

arXiv:1412.5648•Journal of High Energy Physics•May 1, 2015•92 citations

Dynamics of entanglement entropy from Einstein equation

Masahiro Nozaki A, Tokiro Numasawa, Andrea Prudenziati +1 more

We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using anti-de Sitter/conformal field theory (AdS/CFT). This is aimed at a first step to finding a counterpart of the Einstein equation in CFT language. In particular, we point out that the entanglement entropy satisfies differential equations that 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.

arXiv:1304.7100•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 18, 2013•116 citations
The of the entanglement entropies between disjoint subsystems enables us to show that the dynamical equation of the entanglement entropy in CFT 2 is precisely the three-dimensional Einstein equation. We establish a profound relation between the cosmological constant and CFT 2 entanglement entropy. Thus, entanglement entropies induce internal gravitational geometries in CFT 2 . Extracting the dual metric from an entanglement entropy becomes a straightforward procedure. Remarkably, we discover that the renormalization group equation is a geometric identity.
In this thesis we review how entanglement entropy is related to the gravitational equations of motion by means of the AdS/CFT correspondence. We consider the entanglement entropy of a ball-shaped spatial region in the CFT vacuum and derive a holographic relation between the relative entropy and the equations of motion in the dual bulk spacetime. For small perturbations around the CFT vacuum, we show that the first law of entanglement entropy requires the dual metric to satisfy the linearized Einstein equation when the entanglement entropy is given by the area of an extremal surface in the bulk. We discuss a recent paper in which the complete Einstein equation is derived and explain why we believe this result is incomplete. This thesis serves both as a commentary on this paper, as well as a detailed review of the relationship between entanglement and gravity.
Abstract In the context of dS3/CFT2, we propose a timelike entanglement entropy defined by the renormalization group flow. This timelike entanglement entropy is calculated in CFT by using the Callan-Symanzik equation. We find an exact match between this entanglement entropy and the length of a timelike geodesic connecting two different spacelike surfaces in dS3. The counterpart of this entanglement entropy in AdS3 is a spacelike one, also induced by RG flow and extends all the way into the bulk of AdS3. As a result, in both AdS3/CFT2 and dS3/CFT2, there exist exactly three entanglement entropies, providing precisely sufficient information to reconstruct the three-dimensional bulk geometry.
We explicitly show how to derive the (D+1)(D+1)(D+1)-dimensional Einstein equation from the entanglement entropy between codimension-one {\it disjoint} regions in DDD-dimensional conformal field theory.
We further develop perturbative methods used to calculate entanglement entropy (EE) away from an interacting CFT fixed point. At second order we find certain universal terms in the renormalized EE which were predicted previously from holography and which we find hold universally for relevant deformations of any CFT in any dimension. We use both replica methods and direct methods to calculate the EE and in both cases find a non-local integral expression involving the CFT two point function. We show that this integral expression can be written as a local integral over a higher dimensional bulk modular hamiltonian in an emergent AdS space-time. This bulk modular hamiltonian is associated to an emergent scalar field dual to the deforming operator. We generalize to arbitrary spatially dependent couplings where a linearized metric emerges naturally as a way of efficiently encoding the field theory entanglement: by demanding that Einstein’s equations coupled to the bulk scalar field are satisfied, we show that EE can be calculated as the area of this metric. Not only does this show a direct emergence of a higher dimensional gravitational theory from any CFT, it allows for effective evaluation of the the integrals required to calculate EE perturbativly. Our results can also be interpreted as relating the non-locality of the modular hamiltonian for a spherical region in non-CFTs and the non-locality of the holographic bulk to boundary map.
We study the dynamics of entanglement entropy for weakly excited states in conformal field theories by using anti-de Sitter/conformal field theory (AdS/CFT). This is aimed at a first step to finding a counterpart of the Einstein equation in CFT language. In particular, we point out that the entanglement entropy satisfies differential equations that 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.