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:2109.02996 · The European Physical Journal C

Mutual information, islands in black holes and the Page curve

Ashis Saha, Sunandan Gangopadhyay, Jyoti Prasad Saha

Published May 1, 2022• 34 citations
arXivHTMLPDF

Abstract

Abstract The role played by the mutual information of subsystems on the Page curve is explored in this paper. With the total system consisting of the black hole and radiation, together with the inclusion of island, we observe that the vanishing of mutual information between $$B_+$$ B + and $$B_-$$ B - which in turn means the disconnected phase of the entanglement wedge corresponding to $$B_+\cup B_-$$ B + ∪ B - , yields a time scale of the order of scrambling time. This results in a time independent expression for the fine grained entropy of Hawking radiation consistent with the correct Page curve. We also find corrections to this entropy and Page time which are logarithmic and inverse power law in form.

DOI
10.1140/epjc/s10052-022-10426-6

Related papers

Role of mutual information in the Page curve

Anirban Roy Chowdhury, Ashis Saha, Sunandan Gangopadhyay

In this work, we give two proposals regarding the status of connectivity of entanglement wedges and the associated saturation of mutual information. The first proposal has been given for the scenario before the Page time depicting the fact that at a particular value of the observer's time ${t}_{b}={t}_{R}$ (where ${t}_{R}\ensuremath{\ll}\ensuremath{\beta}$), the mutual information $I({R}_{+}:{R}_{\ensuremath{-}})$ vanishes representing the disconnected phase of the radiation entanglement wedge. We argue that this time is the Hartman-Maldacena time at which the fine-grained entropy of radiation goes as $S(R)\ensuremath{\sim}\mathrm{log}(\ensuremath{\beta})$, where $\ensuremath{\beta}$ is the inverse of Hawking temperature of the black hole. On the other hand, the second proposal probes the crucial role played by the mutual information of black hole subsystems in obtaining the correct Page curve of radiation.

arXiv:2207.13029•Physical review. D/Physical review. D.•Oct 27, 2022•27 citations

New insights on mutual information in the island approach to the Page curve

Anirban Roy Chowdhury, Souvik Paul, Sunandan Gangopadhyay

Abstract In this article, we have presented one of the very important observations, regarding the behavior of mutual information of two different sets of subsystems in the after Page time scenario. This provides us with some deep insights about the redistribution of geometrical correlation between the two different sets of subsystems on a Cauchy slice. In our earlier works, we have shown how the saturation of mutual information between two specific subsystems plays a vital role in obtaining the correct Page curve for the eternal black hole. In those works, we have shown that the mutual information between $$B_{+}$$ B + and $$B_{-}$$ B - , that is, $$I(B_{+}: B_{-})$$ I ( B + : B - ) , vanishes at scrambling time, which leads to the correct Page curve. That means that at scrambling time, there is no correlation between $$B_{+}$$ B + and $$B_{-}$$ B - . Remarkably, it is observed that at this particular value of observer’s time, the mutual information between $${\mathcal {I}}$$ I and R , that is, $$I({\mathcal {I}}:R)$$ I ( I : R ) , becomes singular. This indicates that the regions $${\mathcal {I}}$$ I and R become maximally entangled. This provides us with a notion of the transfer of geometric correlation between different regions on the Cauchy slice. In this work, we have also provided a way to calculate the tripartite mutual information of regions $${\mathcal {I}}$$ I , $$R_{+}$$ R + and $$R_{-}$$ R - , that is, $$I({\mathcal {I}}:R_{+}{:}R_{-})$$ I ( I : R + : R - ) on the Cauchy slice using the earlier results involving the bipartite regions. This is a new result which was missing in the earlier literature.

arXiv:2607.04706•The European Physical Journal C•Sep 2, 2026•0 citations

Mutual information of subsystems and the Page curve for the Schwarzschild–de Sitter black hole

Anirban Roy Chowdhury, Ashis Saha, Sunandan Gangopadhyay

In this work, we show that the two proposals associated with the mutual information of matter fields can be given for an eternal Schwarzschild black hole in de Sitter spacetime. These proposals also depict the status of associated entanglement wedges and their role-play in obtaining the correct Page curve of radiation. The first proposal has been given for the before Page time scenario, which shows that the mutual information $I({R}_{H}^{+}:{R}_{H}^{\ensuremath{-}})$ vanishes at a certain value of the observer's time ${t}_{{b}_{H}}={t}_{H}$ (where ${t}_{H}\ensuremath{\ll}{\ensuremath{\beta}}_{H}$). We claim that this is the Hartman-Maldacena time at which the entanglement wedge associated with ${R}_{H}^{+}\ensuremath{\cup}{R}_{H}^{\ensuremath{-}}$ gets disconnected and the fine-grained radiation entropy has the form $S({R}_{H})\ensuremath{\sim}\mathrm{log}({\ensuremath{\beta}}_{H})$. The second proposal depicts the fact that just after the Page time, when the replica wormholes are the dominating saddle points, the mutual information $I({B}_{H}^{+}:{B}_{H}^{\ensuremath{-}})$ vanishes as soon as the time difference ${t}_{{a}_{H}}\ensuremath{-}{t}_{{b}_{H}}$ equals the scrambling time. Holographically, this reflects that the entanglement wedge associated with ${B}_{H}^{+}\ensuremath{\cup}{B}_{H}^{\ensuremath{-}}$ jumps to the disconnected phase at this particular timescale. Furthermore, these two proposals lead us to the correct time evolution of the fine-grained entropy of radiation as portrayed by the Page curve. We have also shown that similar observations can be obtained for the radiation associated with the cosmological horizon.

arXiv:2303.14062•Physical review. D/Physical review. D.•Jul 6, 2023•19 citations

Effect of backreaction on island, Page curve and mutual information

Parul Jain, Sanjay Pant, Himanshu Parihar

We compute the entanglement entropy of Hawking radiation in a bath attached to a deformed eternal AdS black hole. This black hole is dual to the two identical strongly coupled large- N c thermal field theories, where each theory is backreacted (deformed) by the presence of a uniform static distribution of heavy fundamental quarks. In our observation we find that the entanglement entropy of Hawking radiation increases in a quadratic manner for an early time and linearly for the late time. The large time expression for the entanglement entropy of Hawking radiation is used to find the Page curve and Page time. After the Page time, the entanglement entropy saturates to a constant value due to the appearance of an island. We observe that introducing deformation (backreaction) delays the appearance of island and shifts the Page curve to a later time. Subsequently, the computation of the scrambling time reveals an increase with the backreaction parameter, suggesting a longer duration for information retrieval in the presence of deformation. Moreover, our analysis of the mutual information between the radiation subsystems shows that it vanishes at a critical time which increases with the deformation before the Page time. After the Page time, the appearance of island leads to the vanishing of mutual information between black hole subsystems and gives the time difference of the order of scrambling time.

arXiv:2311.08186•Nuclear Physics B•Jun 20, 2025•3 citations

Entanglement Islands and Thermodynamics of the Black Hole in Asymptotically Safe Quantum Gravity

Sobhan Kazempour, Sichun Sun, Chengye Yu

We study thermodynamic properties and the entanglement island of a black hole in asymptotically safe quantum gravity, analyzing key thermodynamic quantities such as the Hawking temperature, heat capacity, and entropy, as well as the mass-horizon radius relation. Unlike Schwarzschild black holes, the temperature decreases with mass near the evaporation endpoint, signaling a phase transition and possible stable remnant. The entanglement entropy of Hawking radiation is obtained both with and without island contributions. Without islands, the radiation entropy grows linearly indefinitely, leading to the information paradox. By including island contributions and extremizing the generalized entropy functional, we resolve this paradox. At late times, the radiation entropy saturates at the Bekenstein-Hawking entropy, confirming unitary evolution. From this, we derive the Page time and scrambling time by equating early- and late-time entanglement entropies. The result of this study establishes the finiteness of the radiation entropy and consistency with quantum mechanics.

gr-qcCross-listedarXiv:2512.21995•Dec 29, 2025

The Page curve from the entanglement membrane

Mike Blake, Anthony P. Thompson

Abstract We study entanglement dynamics in toy models of black hole information built out of chaotic many-body quantum systems, by utilising a coarse-grained description of entanglement dynamics in such systems known as the ‘entanglement membrane’. We show that in these models the Page curve associated to the entropy of Hawking radiation arises from a transition in the entanglement membrane around the Page time, in an analogous manner to the change in quantum extremal surfaces that leads to the Page curve in semi-classical gravity. We also use the entanglement membrane prescription to study the Hayden-Preskill protocol, and demonstrate how information initially encoded in the black hole is rapidly transferred to the radiation around the Page time. Our results relate recent developments in black hole information to generic features of entanglement dynamics in chaotic many-body quantum systems.

arXiv:2306.13140•Journal of High Energy Physics•Nov 2, 2023•3 citations
Abstract The role played by the mutual information of subsystems on the Page curve is explored in this paper. With the total system consisting of the black hole and radiation, together with the inclusion of island, we observe that the vanishing of mutual information between B+B_+B+​ B + and B−B_-B−​ B - which in turn means the disconnected phase of the entanglement wedge corresponding to B+∪B−B_+\cup B_-B+​∪B−​ B + ∪ B - , yields a time scale of the order of scrambling time. This results in a time independent expression for the fine grained entropy of Hawking radiation consistent with the correct Page curve. We also find corrections to this entropy and Page time which are logarithmic and inverse power law in form.
In this work, we give two proposals regarding the status of connectivity of entanglement wedges and the associated saturation of mutual information. The first proposal has been given for the scenario before the Page time depicting the fact that at a particular value of the observer's time tb=tR{t}_{b}={t}_{R}tb​=tR​ (where ${t}_{R}\ensuremath{\ll}\ensuremath{\beta}$), the mutual information $I({R}_{+}:{R}_{\ensuremath{-}})$ vanishes representing the disconnected phase of the radiation entanglement wedge. We argue that this time is the Hartman-Maldacena time at which the fine-grained entropy of radiation goes as $S(R)\ensuremath{\sim}\mathrm{log}(\ensuremath{\beta})$, where $\ensuremath{\beta}$ is the inverse of Hawking temperature of the black hole. On the other hand, the second proposal probes the crucial role played by the mutual information of black hole subsystems in obtaining the correct Page curve of radiation.
Abstract In this article, we have presented one of the very important observations, regarding the behavior of mutual information of two different sets of subsystems in the after Page time scenario. This provides us with some deep insights about the redistribution of geometrical correlation between the two different sets of subsystems on a Cauchy slice. In our earlier works, we have shown how the saturation of mutual information between two specific subsystems plays a vital role in obtaining the correct Page curve for the eternal black hole. In those works, we have shown that the mutual information between B+B_{+}B+​ B + and B−B_{-}B−​ B - , that is, I(B+:B−)I(B_{+}: B_{-})I(B+​:B−​) I ( B + : B - ) , vanishes at scrambling time, which leads to the correct Page curve. That means that at scrambling time, there is no correlation between B+B_{+}B+​ B + and B−B_{-}B−​ B - . Remarkably, it is observed that at this particular value of observer’s time, the mutual information between I{\mathcal {I}}I I and R , that is, I(I:R)I({\mathcal {I}}:R)I(I:R) I ( I : R ) , becomes singular. This indicates that the regions I{\mathcal {I}}I I and R become maximally entangled. This provides us with a notion of the transfer of geometric correlation between different regions on the Cauchy slice. In this work, we have also provided a way to calculate the tripartite mutual information of regions I{\mathcal {I}}I I , R+R_{+}R+​ R + and R−R_{-}R−​ R - , that is, I(I:R+:R−)I({\mathcal {I}}:R_{+}{:}R_{-})I(I:R+​:R−​) I ( I : R + : R - ) on the Cauchy slice using the earlier results involving the bipartite regions. This is a new result which was missing in the earlier literature.
In this work, we show that the two proposals associated with the mutual information of matter fields can be given for an eternal Schwarzschild black hole in de Sitter spacetime. These proposals also depict the status of associated entanglement wedges and their role-play in obtaining the correct Page curve of radiation. The first proposal has been given for the before Page time scenario, which shows that the mutual information $I({R}_{H}^{+}:{R}_{H}^{\ensuremath{-}})$ vanishes at a certain value of the observer's time tbH=tH{t}_{{b}_{H}}={t}_{H}tbH​​=tH​ (where ${t}_{H}\ensuremath{\ll}{\ensuremath{\beta}}_{H}$). We claim that this is the Hartman-Maldacena time at which the entanglement wedge associated with ${R}_{H}^{+}\ensuremath{\cup}{R}_{H}^{\ensuremath{-}}$ gets disconnected and the fine-grained radiation entropy has the form $S({R}_{H})\ensuremath{\sim}\mathrm{log}({\ensuremath{\beta}}_{H})$. The second proposal depicts the fact that just after the Page time, when the replica wormholes are the dominating saddle points, the mutual information $I({B}_{H}^{+}:{B}_{H}^{\ensuremath{-}})$ vanishes as soon as the time difference ${t}_{{a}_{H}}\ensuremath{-}{t}_{{b}_{H}}$ equals the scrambling time. Holographically, this reflects that the entanglement wedge associated with ${B}_{H}^{+}\ensuremath{\cup}{B}_{H}^{\ensuremath{-}}$ jumps to the disconnected phase at this particular timescale. Furthermore, these two proposals lead us to the correct time evolution of the fine-grained entropy of radiation as portrayed by the Page curve. We have also shown that similar observations can be obtained for the radiation associated with the cosmological horizon.
We compute the entanglement entropy of Hawking radiation in a bath attached to a deformed eternal AdS black hole. This black hole is dual to the two identical strongly coupled large- N c thermal field theories, where each theory is backreacted (deformed) by the presence of a uniform static distribution of heavy fundamental quarks. In our observation we find that the entanglement entropy of Hawking radiation increases in a quadratic manner for an early time and linearly for the late time. The large time expression for the entanglement entropy of Hawking radiation is used to find the Page curve and Page time. After the Page time, the entanglement entropy saturates to a constant value due to the appearance of an island. We observe that introducing deformation (backreaction) delays the appearance of island and shifts the Page curve to a later time. Subsequently, the computation of the scrambling time reveals an increase with the backreaction parameter, suggesting a longer duration for information retrieval in the presence of deformation. Moreover, our analysis of the mutual information between the radiation subsystems shows that it vanishes at a critical time which increases with the deformation before the Page time. After the Page time, the appearance of island leads to the vanishing of mutual information between black hole subsystems and gives the time difference of the order of scrambling time.
We study thermodynamic properties and the entanglement island of a black hole in asymptotically safe quantum gravity, analyzing key thermodynamic quantities such as the Hawking temperature, heat capacity, and entropy, as well as the mass-horizon radius relation. Unlike Schwarzschild black holes, the temperature decreases with mass near the evaporation endpoint, signaling a phase transition and possible stable remnant. The entanglement entropy of Hawking radiation is obtained both with and without island contributions. Without islands, the radiation entropy grows linearly indefinitely, leading to the information paradox. By including island contributions and extremizing the generalized entropy functional, we resolve this paradox. At late times, the radiation entropy saturates at the Bekenstein-Hawking entropy, confirming unitary evolution. From this, we derive the Page time and scrambling time by equating early- and late-time entanglement entropies. The result of this study establishes the finiteness of the radiation entropy and consistency with quantum mechanics.
Abstract We study entanglement dynamics in toy models of black hole information built out of chaotic many-body quantum systems, by utilising a coarse-grained description of entanglement dynamics in such systems known as the ‘entanglement membrane’. We show that in these models the Page curve associated to the entropy of Hawking radiation arises from a transition in the entanglement membrane around the Page time, in an analogous manner to the change in quantum extremal surfaces that leads to the Page curve in semi-classical gravity. We also use the entanglement membrane prescription to study the Hayden-Preskill protocol, and demonstrate how information initially encoded in the black hole is rapidly transferred to the radiation around the Page time. Our results relate recent developments in black hole information to generic features of entanglement dynamics in chaotic many-body quantum systems.