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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
07 November 2022
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP11(2022)024
References
S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
S.W. Hawking, Particle creation by black holes, in Euclidean quantum gravity, World Scientific (1975), pp. 167–188 [DOI].
S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
D.N. Page, Hawking radiation and black hole thermodynamics, New J. Phys. 7 (2005) 203 [hep-th/0409024] [INSPIRE].
D.N. Page, Time Dependence of Hawking Radiation Entropy, JCAP 09 (2013) 028 [arXiv:1301.4995] [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The Stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
D. Harlow and P. Hayden, Quantum computation vs. firewalls, JHEP 06 (2013) 085.
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
G. Penington, Entanglement wedge reconstruction and the information paradox, JHEP 09 (2020) 002.
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063.
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [arXiv:2006.06872] [INSPIRE].
H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, Information Flow in Black Hole Evaporation, JHEP 03 (2020) 152 [arXiv:1911.03402] [INSPIRE].
M. Alishahiha, A. Faraji Astaneh and A. Naseh, Island in the presence of higher derivative terms, JHEP 02 (2021) 035 [arXiv:2005.08715] [INSPIRE].
K. Hashimoto, N. Iizuka and Y. Matsuo, Islands in Schwarzschild black holes, JHEP 06 (2020) 085 [arXiv:2004.05863] [INSPIRE].
T. Anegawa and N. Iizuka, Notes on islands in asymptotically flat 2d dilaton black holes, JHEP 07 (2020) 036.
T. Hartman, E. Shaghoulian and A. Strominger, Islands in asymptotically flat 2d gravity, JHEP 07 (2020) 022.
H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, Evaporating black holes coupled to a thermal bath, JHEP 01 (2021) 065.
A. Bhattacharya, A. Chanda, S. Maulik, C. Northe and S. Roy, Topological shadows and complexity of islands in multiboundary wormholes, JHEP 02 (2021) 152.
F. Deng, J. Chu and Y. Zhou, Defect extremal surface as the holographic counterpart of island formula, JHEP 03 (2021) 008.
X. Wang, R. Li and J. Wang, Islands and page curves of reissner-nordström black holes, JHEP 04 (2021) 103.
S. He, Y. Sun, L. Zhao and Y.-X. Zhang, The universality of islands outside the horizon, JHEP 05 (2022) 047 [arXiv:2110.07598] [INSPIRE].
F.F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, Page curve for an evaporating black hole, JHEP 05 (2020) 091.
C. Krishnan, V. Patil and J. Pereira, Page Curve and the Information Paradox in Flat Space, arXiv:2005.02993 [INSPIRE].
W. Sybesma, Pure de Sitter space and the island moving back in time, Class. Quant. Grav. 38 (2021) 145012 [arXiv:2008.07994] [INSPIRE].
C.-J. Chou, H.B. Lao and Y. Yang, Page Curve of Effective Hawking Radiation, arXiv:2111.14551 [INSPIRE].
T.J. Hollowood, S.P. Kumar, A. Legramandi and N. Talwar, Grey-body factors, irreversibility and multiple island saddles, JHEP 03 (2022) 110 [arXiv:2111.02248] [INSPIRE].
K. Suzuki and T. Takayanagi, BCFT and Islands in two dimensions, JHEP 06 (2022) 095 [arXiv:2202.08462] [INSPIRE].
Y.-k. Suzuki and S. Terashima, On the Dynamics in the AdS/BCFT Correspondence, arXiv:2205.10600 [INSPIRE].
A. Bhattacharya, A. Bhattacharyya, P. Nandy and A.K. Patra, Bath deformations, islands, and holographic complexity, Phys. Rev. D 105 (2022) 066019 [arXiv:2112.06967] [INSPIRE].
A. Bhattacharya, A. Bhattacharyya, P. Nandy and A.K. Patra, Partial islands and subregion complexity in geometric secret-sharing model, JHEP 12 (2021) 091 [arXiv:2109.07842] [INSPIRE].
E. Caceres, A. Kundu, A.K. Patra and S. Shashi, Page curves and bath deformations, SciPost Phys. Core 5 (2022) 033 [arXiv:2107.00022] [INSPIRE].
A. Bhattacharya, A. Bhattacharyya, P. Nandy and A.K. Patra, Islands and complexity of eternal black hole and radiation subsystems for a doubly holographic model, JHEP 05 (2021) 135 [arXiv:2103.15852] [INSPIRE].
E. Caceres, A. Kundu, A.K. Patra and S. Shashi, Warped information and entanglement islands in AdS/WCFT, JHEP 07 (2021) 004 [arXiv:2012.05425] [INSPIRE].
A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [INSPIRE].
A. Almheiri, R. Mahajan and J.E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020) 001 [arXiv:1911.09666] [INSPIRE].
T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].
C.-S. Chu and R.-X. Miao, Anomalous Transport in Holographic Boundary Conformal Field Theories, JHEP 07 (2018) 005 [arXiv:1804.01648] [INSPIRE].
R.-X. Miao, Holographic BCFT with Dirichlet Boundary Condition, JHEP 02 (2019) 025 [arXiv:1806.10777] [INSPIRE].
Y. Ling, Y. Liu and Z.-Y. Xian, Island in Charged Black Holes, JHEP 03 (2021) 251 [arXiv:2010.00037] [INSPIRE].
H.Z. Chen, R.C. Myers, D. Neuenfeld, I.A. Reyes and J. Sandor, Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane, JHEP 10 (2020) 166 [arXiv:2006.04851] [INSPIRE].
H. Geng et al., Information Transfer with a Gravitating Bath, SciPost Phys. 10 (2021) 103 [arXiv:2012.04671] [INSPIRE].
H. Geng and A. Karch, Massive islands, JHEP 09 (2020) 121 [arXiv:2006.02438] [INSPIRE].
C. Krishnan, Critical Islands, JHEP 01 (2021) 179 [arXiv:2007.06551] [INSPIRE].
H.Z. Chen, R.C. Myers, D. Neuenfeld, I.A. Reyes and J. Sandor, Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane, JHEP 12 (2020) 025 [arXiv:2010.00018] [INSPIRE].
J. Hernandez, R.C. Myers and S.-M. Ruan, Quantum extremal islands made easy. Part III. Complexity on the brane, JHEP 02 (2021) 173 [arXiv:2010.16398] [INSPIRE].
G. Grimaldi, J. Hernandez and R.C. Myers, Quantum extremal islands made easy. Part IV. Massive black holes on the brane, JHEP 03 (2022) 136 [arXiv:2202.00679] [INSPIRE].
R.-X. Miao, An Exact Construction of Codimension two Holography, JHEP 01 (2021) 150 [arXiv:2009.06263] [INSPIRE].
I. Akal, Y. Kusuki, T. Takayanagi and Z. Wei, Codimension two holography for wedges, Phys. Rev. D 102 (2020) 126007 [arXiv:2007.06800] [INSPIRE].
I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi and Z. Wei, Entanglement Entropy in a Holographic Moving Mirror and the Page Curve, Phys. Rev. Lett. 126 (2021) 061604 [arXiv:2011.12005] [INSPIRE].
F. Omidi, Entropy of Hawking radiation for two-sided hyperscaling violating black branes, JHEP 04 (2022) 022 [arXiv:2112.05890] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell and D. Wakeham, Information radiation in bcft models of black holes, JHEP 05 (2020) 004.
A. Karlsson, Replica wormhole and island incompatibility with monogamy of entanglement, arXiv:2007.10523 [INSPIRE].
V. Balasubramanian, A. Kar and T. Ugajin, Islands in de Sitter space, JHEP 02 (2021) 072.
V. Balasubramanian, A. Kar and T. Ugajin, Entanglement between two gravitating universes, Class. Quant. Grav. 1 (2021) 1.
V. Balasubramanian, A. Kar and T. Ugajin, Entanglement between two disjoint universes, JHEP 02 (2021) 136.
A. Miyata and T. Ugajin, Evaporation of black holes in flat space entangled with an auxiliary universe, PTEP 2022 (2022) 013B13 [arXiv:2104.00183] [INSPIRE].
A. Miyata and T. Ugajin, Entanglement between two evaporating black holes, JHEP 09 (2022) 009 [arXiv:2111.11688] [INSPIRE].
D. Marolf and H. Maxfield, Observations of hawking radiation: the page curve and baby universes, JHEP 04 (2021) 272.
D. Marolf and H. Maxfield, Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information, JHEP 08 (2020) 044.
V. Balasubramanian, A. Kar, S.F. Ross and T. Ugajin, Spin structures and baby universes, JHEP 09 (2020) 192.
C. Peng, J. Tian and J. Yu, Baby universes, ensemble averages and factorizations with matters, arXiv:2111.14856 [INSPIRE].
R. Renner and J. Wang, The black hole information puzzle and the quantum de Finetti theorem, arXiv:2110.14653 [INSPIRE].
T. Li, J. Chu and Y. Zhou, Reflected entropy for an evaporating black hole, JHEP 11 (2020) 155.
J. Kumar Basak, D. Basu, V. Malvimat, H. Parihar and G. Sengupta, Islands for entanglement negativity, SciPost Phys. 12 (2022) 003 [arXiv:2012.03983] [INSPIRE].
K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, Probing Hawking radiation through capacity of entanglement, JHEP 05 (2021) 062 [arXiv:2102.02425] [INSPIRE].
K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, Replica wormholes and capacity of entanglement, JHEP 10 (2021) 227.
S. Vardhan, J. Kudler-Flam, H. Shapourian and H. Liu, Mixed-state entanglement and information recovery in thermalized states and evaporating black holes, arXiv:2112.00020 [INSPIRE].
I. Akal, T. Kawamoto, S.-M. Ruan, T. Takayanagi and Z. Wei, Page curve under final state projection, Phys. Rev. D 105 (2022) 126026 [arXiv:2112.08433] [INSPIRE].
Y. Ling, P. Liu, Y. Liu, C. Niu, Z.-Y. Xian and C.-Y. Zhang, Reflected entropy in double holography, JHEP 02 (2022) 037 [arXiv:2109.09243] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
Y. Gu, A. Lucas and X.-L. Qi, Spread of entanglement in a Sachdev-Ye-Kitaev chain, JHEP 09 (2017) 120 [arXiv:1708.00871] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
V. Balasubramanian, A. Kar, O. Parrikar, G. Sárosi and T. Ugajin, Geometric secret sharing in a model of Hawking radiation, JHEP 01 (2021) 177 [arXiv:2003.05448] [INSPIRE].
L. Anderson, O. Parrikar and R.M. Soni, Islands with gravitating baths: towards ER=EPR, JHEP 10 (2021) 226.
N. Engelhardt and r. Folkestad, Canonical purification of evaporating black holes, Phys. Rev. D 105 (2022) 086010 [arXiv:2201.08395] [INSPIRE].
V. Balasubramanian, B. Craps, M. Khramtsov and E. Shaghoulian, Submerging islands through thermalization, JHEP 10 (2021) 048 [arXiv:2107.14746] [INSPIRE].
G.R. Dvali, G. Gabadadze and M. Porrati, 4-D gravity on a brane in 5-D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].
Y. Chen, X.-L. Qi and P. Zhang, Replica wormhole and information retrieval in the SYK model coupled to Majorana chains, JHEP 06 (2020) 121 [arXiv:2003.13147] [INSPIRE].
H. Geng, S. Lüst, R.K. Mishra and D. Wakeham, Holographic bcfts and communicating black holes, JHEP 08 (2021) 003.
H. Geng et al., Entanglement phase structure of a holographic BCFT in a black hole background, JHEP 05 (2022) 153 [arXiv:2112.09132] [INSPIRE].
M. Afrasiar, J. Kumar Basak, A. Chandra and G. Sengupta, Islands for Entanglement Negativity in Communicating Black Holes, arXiv:2205.07903 [INSPIRE].
L. Randall and R. Sundrum, An Alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [INSPIRE].
A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008 [hep-th/0011156] [INSPIRE].
M. Nozaki, T. Takayanagi and T. Ugajin, Central Charges for BCFTs and Holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].
O.J.C. Dias, J.E. Santos and B. Way, Numerical Methods for Finding Stationary Gravitational Solutions, Class. Quant. Grav. 33 (2016) 133001 [arXiv:1510.02804] [INSPIRE].
P. Figueras, J. Lucietti and T. Wiseman, Ricci solitons, Ricci flow, and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua, Class. Quant. Grav. 28 (2011) 215018 [arXiv:1104.4489] [INSPIRE].
P. Figueras and T. Wiseman, On the existence of stationary Ricci solitons, Class. Quant. Grav. 34 (2017) 145007 [arXiv:1610.06178] [INSPIRE].
D.-C. Dai, D. Minic, D. Stojkovic and C. Fu, Testing the ER = EPR conjecture, Phys. Rev. D 102 (2020) 066004 [arXiv:2002.08178] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
P. Gao, D.L. Jafferis and A.C. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
Y. Chen and P. Zhang, Entanglement Entropy of Two Coupled SYK Models and Eternal Traversable Wormhole, JHEP 07 (2019) 033 [arXiv:1903.10532] [INSPIRE].
P. Figueras and T. Wiseman, Stationary holographic plasma quenches and numerical methods for non-Killing horizons, Phys. Rev. Lett. 110 (2013) 171602 [arXiv:1212.4498] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
P. Saad, S. Shenker and S. Yao, Comments on wormholes and factorization, arXiv:2107.13130 [INSPIRE].
P. Saad, S.H. Shenker, D. Stanford and S. Yao, Wormholes without averaging, arXiv:2103.16754 [INSPIRE].
B. Mukhametzhanov, Half-wormholes in SYK with one time point, SciPost Phys. 12 (2022) 029 [arXiv:2105.08207] [INSPIRE].
B. Mukhametzhanov, Factorization and complex couplings in SYK and in Matrix Models, arXiv:2110.06221 [INSPIRE].
C. Peng, J. Tian and Y. Yang, Half-Wormholes and Ensemble Averages, arXiv:2205.01288 [INSPIRE].
F. Alet, M. Hanada, A. Jevicki and C. Peng, Entanglement and Confinement in Coupled Quantum Systems, JHEP 02 (2021) 034 [arXiv:2001.03158] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2205.14596
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Liu, Y., Xian, ZY., Peng, C. et al. Black holes entangled by radiation. J. High Energ. Phys. 2022, 179 (2022). https://doi.org/10.1007/JHEP09(2022)179
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2022)179