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arXiv:2303.14062 · Physical review. D/Physical review. D.

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

Anirban Roy Chowdhury, Ashis Saha, Sunandan Gangopadhyay

Published Jul 6, 2023• 19 citations
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Abstract

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.

DOI
10.1103/physrevd.108.026003

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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(RH+:RH\ensuremath−)I({R}_{H}^{+}:{R}_{H}^{\ensuremath{-}})I(RH+​:RH\ensuremath−​) vanishes at a certain value of the observer's time tbH=tH{t}_{{b}_{H}}={t}_{H}tbH​​=tH​ (where tH\ensuremath≪\ensuremathβH{t}_{H}\ensuremath{\ll}{\ensuremath{\beta}}_{H}tH​\ensuremath≪\ensuremathβH​). We claim that this is the Hartman-Maldacena time at which the entanglement wedge associated with RH+\ensuremath∪RH\ensuremath−{R}_{H}^{+}\ensuremath{\cup}{R}_{H}^{\ensuremath{-}}RH+​\ensuremath∪RH\ensuremath−​ gets disconnected and the fine-grained radiation entropy has the form S(RH)\ensuremath∼log(\ensuremathβH)S({R}_{H})\ensuremath{\sim}\mathrm{log}({\ensuremath{\beta}}_{H})S(RH​)\ensuremath∼log(\ensuremathβH​). 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(BH+:BH\ensuremath−)I({B}_{H}^{+}:{B}_{H}^{\ensuremath{-}})I(BH+​:BH\ensuremath−​) vanishes as soon as the time difference taH\ensuremath−tbH{t}_{{a}_{H}}\ensuremath{-}{t}_{{b}_{H}}taH​​\ensuremath−tbH​​ equals the scrambling time. Holographically, this reflects that the entanglement wedge associated with BH+\ensuremath∪BH\ensuremath−{B}_{H}^{+}\ensuremath{\cup}{B}_{H}^{\ensuremath{-}}BH+​\ensuremath∪BH\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.
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.
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.
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.
A bstract We investigate the evolution of entanglement within an open, strongly coupled system interacting with a heat bath as its environment, in the frameworks of both the doubly holographic model and the Sachdev-Ye-Kitaev (SYK) model. Generally, the entanglement within the system initially increases as due to internal interactions; however, it eventually dissipates into the environment. In the doubly holographic setup, we consider an end-of-the-world brane in the bulk to represent an eternal black hole coupled with its radiation and the evolution of the global thermofield double (TFD) state. For small black holes, the reflected entropy between the bipartition exhibits a ramp-plateau-slump behavior, where the plateau arises due to the phase transition of the entanglement wedge cross-section before the Page time. Similarly, the mutual information between the bipartition displays a ramp-slop-stabilizing behavior. In quantum mechanics, we consider a double copy of the SYK-plus-bath system in a global TFD state, resembling an eternal black hole interacting with an environment. The Rényi mutual information within the double-copied SYK clusters exhibits a ramp-plateau-slope-stabilizing behavior. The dynamic behaviors of the entanglement quantities observed in these two models are attributable to the competition between the internal interaction of the system and the external interaction with the baths. Our study provides a fine-grained picture of the entanglement dynamics inside black holes before their Page time.
Abstract We construct three models to describe the scenario where two eternal black holes are separated by a flat space, and can eventually be entangled by exchanging radiation. In the doubly holographic setup, we compute the entanglement entropy and mutual information among subsystems and obtain the dynamic phase structure of the entanglement. The formation of entanglement between these two black holes is delayed by the space which the radiation must travel through. If the black holes exchange sufficient Hawking modes, the final state is characterized by a connected entanglement wedge; otherwise, the final entanglement wedge contains two separate islands. In the former case, the entanglement wedge of the black holes forms at the time scale proportional to the size of the flat space between them. While in both cases, the unitarity of the evolution is preserved. When the sizes of the black holes are not equal, we observe a loss of entanglement between the smaller black hole and the radiation at late times. On the field theory side, we consider two Sachdev-Ye-Kitaev (SYK) clusters coupled to a Majorana chain, which resemble two black holes connected by a radiation region. We numerically compute the same entanglement measures and obtain similar phase structures as the bulk results. In general, a time delay of the entanglement between the SYK clusters is found in cases with a long Majorana chain. In particular, when the SYK clusters are different in size, similar entanglement loss between the smaller SYK cluster and the Majorana chain is observed. Finally, we investigate a chain model composed of EPR clusters with particles exchanging between neighboring clusters and reproduce the features of entanglement observed in the previous models.