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

European Physical Society Conference on High Energy Physics (EPS-HEP)

Generalizing the Ryu-Takayanagi formula to probe entanglement shadows of BTZ black holes

Marius Gerbershagen

Published Feb 17, 2022• 0 citations
Publisher

Abstract

We define a generalized entanglement measure in the context of AdS$_3$/CFT$_2$. Compared to the ordinary entanglement entropy between spatial degrees of freedom dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary CFT. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including regions known as entanglement shadows which are not reached by any Ryu-Takayanagi surface. This allows us to describe regions close to the black hole horizon in the one-sided black hole and the wormhole growth in the case of a two-sided black hole from entanglement data in the boundary field theory, leading us to propose that entanglement is in fact enough to reconstruct the full BTZ geometry from boundary data.

DOI
10.22323/1.398.0734

Related papers

Illuminating entanglement shadows of BTZ black holes by a generalized entanglement measure

Marius Gerbershagen

Abstract We define a generalized entanglement measure in the context of the AdS/CFT correspondence. Compared to the ordinary entanglement entropy for a spatial subregion dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary theory. Moreover, we resolve the contribution to the entanglement entropy of strings with different winding numbers in the bulk geometry. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including the entanglement shadow up to the horizon in the one-sided black hole as well as the wormhole growth in the case of a two-sided black hole for an arbitrarily long time. Therefore, we propose that the entanglement structure of the boundary state is enough to reconstruct asymptotically AdS3 geometries up to extremal surface barriers.

arXiv:2105.01097•Journal of High Energy Physics•Oct 22, 2021•9 citations

Entanglement contour from subset entanglement entropies

Qiang Wen

Entanglement contour characterizes the spatial structure of entanglement and quantifies the contribution from the degrees of freedom in any subset of the region $\mathcal{A}$ to the total entanglement entropy $S_{\mathcal{A}}$. Recently in \cite{Wen:2018whg} the author gave a simple proposal for the entanglement contour (or partial entanglement entropy) in two-dimensional theories, which involves the entanglement entropies of all the subsets inside $\mathcal{A}$. In this paper we explicitly study this proposal and show it satisfies many rational requirements for entanglement contour. Together with the holographic picture constructed with the modular planes \cite{Wen:2018whg}, we propose the correspondence between bulk geodesic chords and boundary partial entanglement entropies, which can be considered as a finer version of the Ryu-Takayanagi (RT) formula. As an example we calculate the fine correspondence between the points on $\mathcal{A}$ and the points on the RT surface $\mathcal{E}_{\mathcal{A}}$ for the BTZ black hole. We also give a strategy to extract the local modular flow from our proposal.

•Feb 19, 2019•3 citations

Bulk-boundary entanglement correspondence and the Ryu-Takayanagi conjecture in an AdS2/CFT1 setup

Gabriel Lopes Cardoso, Bernardo Moniz Martins, Suresh Nampuri

A bstract Using recent developments in expressing one-loop partition functions in Euclidean AdS 2 space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in AdS 2 (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global AdS 2 boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the CFT 1 (CQM), each living near one of the two boundaries of global AdS 2 , in a state at global time τ → – ∞. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean AdS 2 disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the AdS 2 / CFT 1 version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in AdS 2 . Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian AdS 2 and show that it precisely matches the boundary entanglement entropy.

arXiv:2502.01144•Journal of High Energy Physics•Jun 19, 2025•3 citations

Pulling the boundary into the bulk

Yasunori Nomura A, Pratik Rath, Nico Salzetta

Motivated by the ability to consistently apply the Ryu-Takayanagi prescription for general convex surfaces and the relationship between entanglement and geometry in tensor networks, we introduce a novel, covariant bulk object---the holographic slice. The holographic slice is found by considering the continual removal of short-range information in a boundary state. It thus provides a natural interpretation as the bulk dual of a series of coarse-grained holographic states. The slice possesses many desirable properties that provide consistency checks for its boundary interpretation. These include the monotonicity of both area and entanglement entropy, uniqueness, and the inability to probe beyond late-time black hole horizons. Additionally, the holographic slice illuminates physics behind entanglement shadows, as minimal-area extremal surfaces anchored to a coarse-grained boundary may probe entanglement shadows. This lets the slice flow through shadows. To aid in developing intuition for these slices, many explicit examples of holographic slices are investigated. Finally, the relationship to tensor networks and renormalization (particularly in $\mathrm{AdS}/\mathrm{CFT}$) is discussed.

arXiv:1805.00523•Physical review. D/Physical review. D.•Jul 6, 2018•50 citations

Entanglement entropy and algebraic holography

Bernard S. Kay

In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.

arXiv:1605.07872•May 25, 2016•0 citations

Holography, entanglement entropy, and conformal field theories with boundaries or defects

Kristan Jensen, Andy O’Bannon

We study entanglement entropy (EE) in conformal field theories (CFTs) in Minkowski space with a planar boundary or with a planar defect of any codimension. In any such boundary CFT or defect CFT (DCFT), we consider the reduced density matrix and associated EE obtained by tracing over the degrees of freedom outside of a (hemi)sphere centered on the boundary or defect. Following Casini, Huerta, and Myers, we map the reduced density matrix to a thermal density matrix of the same theory on hyperbolic space. The EE maps to the thermal entropy of the theory on hyperbolic space. For boundary CFTs and DCFTs dual holographically to Einstein gravity theories, the thermal entropy is equivalent to the Bekenstein--Hawking entropy of a hyperbolic black brane. We show that the horizon of the hyperbolic black brane coincides with the minimal area surface used in Ryu and Takayanagi's conjecture for the holographic calculation of EE. We thus prove their conjecture in these cases. We use our results to compute the R\'enyi entropies and EE in DCFTs in which the defect corresponds to a probe brane in a holographic dual.

arXiv:1309.4523•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 22, 2013•120 citations
We define a generalized entanglement measure in the context of AdS3_33​/CFT2_22​. Compared to the ordinary entanglement entropy between spatial degrees of freedom dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary CFT. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including regions known as entanglement shadows which are not reached by any Ryu-Takayanagi surface. This allows us to describe regions close to the black hole horizon in the one-sided black hole and the wormhole growth in the case of a two-sided black hole from entanglement data in the boundary field theory, leading us to propose that entanglement is in fact enough to reconstruct the full BTZ geometry from boundary data.
Abstract We define a generalized entanglement measure in the context of the AdS/CFT correspondence. Compared to the ordinary entanglement entropy for a spatial subregion dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary theory. Moreover, we resolve the contribution to the entanglement entropy of strings with different winding numbers in the bulk geometry. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including the entanglement shadow up to the horizon in the one-sided black hole as well as the wormhole growth in the case of a two-sided black hole for an arbitrarily long time. Therefore, we propose that the entanglement structure of the boundary state is enough to reconstruct asymptotically AdS3 geometries up to extremal surface barriers.
Entanglement contour characterizes the spatial structure of entanglement and quantifies the contribution from the degrees of freedom in any subset of the region A\mathcal{A}A to the total entanglement entropy SAS_{\mathcal{A}}SA​. Recently in \cite{Wen:2018whg} the author gave a simple proposal for the entanglement contour (or partial entanglement entropy) in two-dimensional theories, which involves the entanglement entropies of all the subsets inside A\mathcal{A}A. In this paper we explicitly study this proposal and show it satisfies many rational requirements for entanglement contour. Together with the holographic picture constructed with the modular planes \cite{Wen:2018whg}, we propose the correspondence between bulk geodesic chords and boundary partial entanglement entropies, which can be considered as a finer version of the Ryu-Takayanagi (RT) formula. As an example we calculate the fine correspondence between the points on A\mathcal{A}A and the points on the RT surface EA\mathcal{E}_{\mathcal{A}}EA​ for the BTZ black hole. We also give a strategy to extract the local modular flow from our proposal.
A bstract Using recent developments in expressing one-loop partition functions in Euclidean AdS 2 space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in AdS 2 (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global AdS 2 boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the CFT 1 (CQM), each living near one of the two boundaries of global AdS 2 , in a state at global time τ → – ∞. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean AdS 2 disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the AdS 2 / CFT 1 version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in AdS 2 . Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian AdS 2 and show that it precisely matches the boundary entanglement entropy.
Motivated by the ability to consistently apply the Ryu-Takayanagi prescription for general convex surfaces and the relationship between entanglement and geometry in tensor networks, we introduce a novel, covariant bulk object---the holographic slice. The holographic slice is found by considering the continual removal of short-range information in a boundary state. It thus provides a natural interpretation as the bulk dual of a series of coarse-grained holographic states. The slice possesses many desirable properties that provide consistency checks for its boundary interpretation. These include the monotonicity of both area and entanglement entropy, uniqueness, and the inability to probe beyond late-time black hole horizons. Additionally, the holographic slice illuminates physics behind entanglement shadows, as minimal-area extremal surfaces anchored to a coarse-grained boundary may probe entanglement shadows. This lets the slice flow through shadows. To aid in developing intuition for these slices, many explicit examples of holographic slices are investigated. Finally, the relationship to tensor networks and renormalization (particularly in AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT) is discussed.
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.
We study entanglement entropy (EE) in conformal field theories (CFTs) in Minkowski space with a planar boundary or with a planar defect of any codimension. In any such boundary CFT or defect CFT (DCFT), we consider the reduced density matrix and associated EE obtained by tracing over the degrees of freedom outside of a (hemi)sphere centered on the boundary or defect. Following Casini, Huerta, and Myers, we map the reduced density matrix to a thermal density matrix of the same theory on hyperbolic space. The EE maps to the thermal entropy of the theory on hyperbolic space. For boundary CFTs and DCFTs dual holographically to Einstein gravity theories, the thermal entropy is equivalent to the Bekenstein--Hawking entropy of a hyperbolic black brane. We show that the horizon of the hyperbolic black brane coincides with the minimal area surface used in Ryu and Takayanagi's conjecture for the holographic calculation of EE. We thus prove their conjecture in these cases. We use our results to compute the R\'enyi entropies and EE in DCFTs in which the defect corresponds to a probe brane in a holographic dual.