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arXiv:2607.04706 · The European Physical Journal C

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

Anirban Roy Chowdhury, Souvik Paul, Sunandan Gangopadhyay

Published Sep 2, 2026• 0 citations
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Abstract

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.

DOI
10.1140/epjc/s10052-026-16291-x

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•Oct 25, 2026•0 citations
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.
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 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.
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.
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 entanglement transfer in a minimal model of two qubits that are coupled with one another through an antiferromagnetic Heisenberg exchange (J~\tilde{J}J~), and where one of them is additionally coupled to a fermionic environment through another antiferromagnetic Heisenberg exchange (JKJ_{K}JK​). By tuning the coupling ratio JK/J~J_{K}/\tilde{J}JK​/J~, the system undergoes a quantum phase transition at T=0T=0T=0, accompanied by a redistribution of entanglement from the d′−dd'-dd′−d qubit-subsystem to the environment. Remarkably, the resulting physics exhibits properties that bear analogy with a quantum-information theoretic perspective of the physics of black hole thermodynamics. Carefully selected bipartite and tripartite mutual information measures displays behaviour analogous to the dynamical evolution of black hole entropy, Hawking entropy, and the Page curve expected during the process of evaporation. An effective temperature scale is obtained from the variation of the ground state energy with respect to changes in the bipartite mutual information between the subsystem and the bath. A steady growth of this temperature with the coupling ratio resembles that of the Hawking temperature with the inverse mass of the black hole. Concomitantly, the emergence of non-Fermi liquid behaviour observed near the quantum critical point and in the strong-coupling phase resembles strange-metal-like physics expected near the event horizon from a holographic duality perspective. Our results establish the minimal model as a platform for studying entanglement transfer and information scrambling within a fully unitary quantum framework, and offer new insights into a resolution of the black hole information paradox.
The core of the black hole information paradox lies in whether Hawking radiation leads to information loss. Based on connectivity engineering, we construct a minimal, gravity-free spin network. By calculating the mutual information and fidelity across different bipartition sizes, we reproduce the complete Page curve. Numerical results show that the peak of mutual information in the geometrically decoupled network (k=3) exhibits a non-trivial scaling behavior with the connectivity k, and reaches saturation at k=3 (with a peak approximately 1.49 times that of the normal network). This not only reveals the reshaping of entanglement capacity by geometric topology but also establishes a topological upper bound for the Page curve peak. Fidelity analysis further reveals that in the late stages of evaporation, the residual subsystem tends towards purification, with the target state fidelity significantly rising to approximately 0.52, which is weakly sensitive to k. This indicates that information is not completely thermalized but exists in a highly compressed global correlation form within the entanglement structure of the black hole residue and external radiation. This model provides a deterministic and strictly reproducible discrete physical benchmark for studying black hole information dynamics.