Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.W. Hawking, Black hole explosions?, Nature 248 (1974) 30.
S.W. Hawking, Particle creation by black holes, in Euclidean quantum gravity, World Scientific (1975), pp. 167–188.
S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [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 [arXiv:1301.4504] [INSPIRE].
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].
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].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [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].
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 et al., The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [arXiv:2006.06872] [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 [arXiv:2004.01601] [INSPIRE].
T. Hartman, E. Shaghoulian and A. Strominger, Islands in Asymptotically Flat 2D Gravity, JHEP 07 (2020) 022 [arXiv:2004.13857] [INSPIRE].
H.Z. Chen et al., Evaporating Black Holes Coupled to a Thermal Bath, JHEP 01 (2021) 065 [arXiv:2007.11658] [INSPIRE].
A. Bhattacharya et al., Topological shadows and complexity of islands in multiboundary wormholes, JHEP 02 (2021) 152 [arXiv:2010.04134] [INSPIRE].
F. Deng, J. Chu and Y. Zhou, Defect extremal surface as the holographic counterpart of Island formula, JHEP 03 (2021) 008 [arXiv:2012.07612] [INSPIRE].
X. Wang, R. Li and J. Wang, Islands and Page curves of Reissner-Nordström black holes, JHEP 04 (2021) 103 [arXiv:2101.06867] [INSPIRE].
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 [arXiv:2004.00598] [INSPIRE].
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, Phys. Rev. D 106 (2022) 066008 [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, JHEP 09 (2022) 103 [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].
Y. Chen, Pulling Out the Island with Modular Flow, JHEP 03 (2020) 033 [arXiv:1912.02210] [INSPIRE].
V. Balasubramanian et al., 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 (2020) 226 [arXiv:2103.14746] [INSPIRE].
V. Balasubramanian, B. Craps, M. Khramtsov and E. Shaghoulian, Submerging islands through thermalization, JHEP 10 (2021) 048 [arXiv:2107.14746] [INSPIRE].
N. Engelhardt and Å. Folkestad, Canonical purification of evaporating black holes, Phys. Rev. D 105 (2022) 086010 [arXiv:2201.08395] [INSPIRE].
C.-J. Chou, H.B. Lao and Y. Yang, Page curve of AdS-Vaidya model for evaporating black holes, JHEP 05 (2024) 342 [arXiv:2306.16744] [INSPIRE].
H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, Entanglement entropy analysis of dyonic black holes using doubly holographic theory, Phys. Rev. D 108 (2023) 126016 [arXiv:2305.18122] [INSPIRE].
B. Ahn et al., Islands in charged linear dilaton black holes, Phys. Rev. D 105 (2022) 046012 [arXiv:2107.07444] [INSPIRE].
A. Karch, H. Sun and C.F. Uhlemann, Double holography in string theory, JHEP 10 (2022) 012 [arXiv:2206.11292] [INSPIRE].
C.F. Uhlemann, Islands and Page curves in 4d from Type IIB, JHEP 08 (2021) 104 [arXiv:2105.00008] [INSPIRE].
A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [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].
H.Z. Chen et al., Information Flow in Black Hole Evaporation, JHEP 03 (2020) 152 [arXiv:1911.03402] [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].
M. Nozaki, T. Takayanagi and T. Ugajin, Central Charges for BCFTs and Holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].
K. Izumi et al., Brane dynamics of holographic BCFTs, JHEP 10 (2022) 050 [arXiv:2205.15500] [INSPIRE].
J. Erdmenger, M. Flory and M.-N. Newrzella, Bending branes for DCFT in two dimensions, JHEP 01 (2015) 058 [arXiv:1410.7811] [INSPIRE].
J. Erdmenger et al., Entanglement Entropy in a Holographic Kondo Model, Fortsch. Phys. 64 (2016) 109 [arXiv:1511.03666] [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 et al., 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 et al., 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 et al., 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].
A. Almheiri et al., Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
M. Rozali et al., Information radiation in BCFT models of black holes, JHEP 05 (2020) 004 [arXiv:1910.12836] [INSPIRE].
A. Karlsson, Replica wormhole and island incompatibility with monogamy of entanglement, arXiv:2007.10523 [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
P. Zhang, Evaporation dynamics of the Sachdev-Ye-Kitaev model, Phys. Rev. B 100 (2019) 245104 [arXiv:1909.10637] [INSPIRE].
A. Almheiri, A. Milekhin and B. Swingle, Universal Constraints on Energy Flow and SYK Thermalization, arXiv:1912.04912 [INSPIRE].
Y. Gu, X.-L. Qi and D. Stanford, Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models, JHEP 05 (2017) 125 [arXiv:1609.07832] [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].
S.-K. Jian and B. Swingle, Note on entropy dynamics in the Brownian SYK model, JHEP 03 (2021) 042 [arXiv:2011.08158] [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].
Y. Liu, Z.-Y. Xian, C. Peng and Y. Ling, Black holes entangled by radiation, JHEP 11 (2022) 043 [Addendum ibid. 11 (2022) 043] [arXiv:2205.14596] [INSPIRE].
P.J. Dodd and J.J. Halliwell, Disentanglement and decoherence by open system dynamics, Phys. Rev. A 69 (2004) 052105.
Z. Sun, X. Wang and C.P. Sun, Disentanglement in a quantum-critical environment, Phys. Rev. A 75 (2007) 062312.
Y.-S. Kim, J.-C. Lee, O. Kwon and Y.-H. Kim, Protecting entanglement from decoherence using weak measurement and quantum measurement reversal, Nature Phys. 8 (2011) 117 [INSPIRE].
A. Del Campo and T. Takayanagi, Decoherence in Conformal Field Theory, JHEP 02 (2020) 170 [arXiv:1911.07861] [INSPIRE].
Y. Yang, Z. Xu and A. del Campo, Decoherence rate in random Lindblad dynamics, Phys. Rev. Res. 6 (2024) 023229 [arXiv:2402.04705] [INSPIRE].
M. Afrasiar, J. Kumar Basak, A. Chandra and G. Sengupta, Islands for entanglement negativity in communicating black holes, Phys. Rev. D 108 (2023) 066013 [arXiv:2205.07903] [INSPIRE].
H. Geng, S. Lüst, R.K. Mishra and D. Wakeham, Holographic BCFTs and Communicating Black Holes, JHEP 08 (2021) 003 [arXiv:2104.07039] [INSPIRE].
H. Geng et al., Entanglement phase structure of a holographic BCFT in a black hole background, JHEP 05 (2022) 153 [arXiv:2112.09132] [INSPIRE].
H. Wang, C. Liu, P. Zhang and A.M. García-García, Entanglement transition and replica wormholes in the dissipative Sachdev-Ye-Kitaev model, Phys. Rev. D 109 (2024) 046005 [arXiv:2306.12571] [INSPIRE].
V. Chandrasekaran, M. Miyaji and P. Rath, Including contributions from entanglement islands to the reflected entropy, Phys. Rev. D 102 (2020) 086009 [arXiv:2006.10754] [INSPIRE].
T. Li, M.-K. Yuan and Y. Zhou, Defect extremal surface for reflected entropy, JHEP 01 (2022) 018 [arXiv:2108.08544] [INSPIRE].
T. Li, J. Chu and Y. Zhou, Reflected Entropy for an Evaporating Black Hole, JHEP 11 (2020) 155 [arXiv:2006.10846] [INSPIRE].
B. Czech, S. Shuai and H. Tang, Entropies and reflected entropies in the Hayden-Preskill protocol, JHEP 02 (2024) 040 [arXiv:2310.16988] [INSPIRE].
L. Randall and R. Sundrum, An alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690 [hep-th/9906064] [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].
S.S. Gubser, AdS / CFT and gravity, Phys. Rev. D 63 (2001) 084017 [hep-th/9912001] [INSPIRE].
R.-X. Miao, C.-S. Chu and W.-Z. Guo, New proposal for a holographic boundary conformal field theory, Phys. Rev. D 96 (2017) 046005 [arXiv:1701.04275] [INSPIRE].
C.-S. Chu, R.-X. Miao and W.-Z. Guo, On New Proposal for Holographic BCFT, JHEP 04 (2017) 089 [arXiv:1701.07202] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
H. Elvang and M. Hadjiantonis, A Practical Approach to the Hamilton-Jacobi Formulation of Holographic Renormalization, JHEP 06 (2016) 046 [arXiv:1603.04485] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178 [arXiv:1905.00577] [INSPIRE].
Y. Ling et al., Reflected entropy in double holography, JHEP 02 (2022) 037 [arXiv:2109.09243] [INSPIRE].
Y. Liu et al., Entanglement of defect subregions in double holography, arXiv:2312.08025 [INSPIRE].
K. Su, P. Zhang and H. Zhai, Page curve from non-Markovianity, JHEP 06 (2020) 156 [arXiv:2101.11238] [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].
Ó.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].
R.A. Janik, J. Jankowski and H. Soltanpanahi, Real-Time dynamics and phase separation in a holographic first order phase transition, Phys. Rev. Lett. 119 (2017) 261601 [arXiv:1704.05387] [INSPIRE].
Q. Chen et al., Critical dynamics in holographic first-order phase transition, JHEP 01 (2023) 056 [arXiv:2209.12789] [INSPIRE].
Q. Chen et al., Quench dynamics in holographic first-order phase transition, Phys. Rev. D 108 (2023) 106017 [arXiv:2211.11291] [INSPIRE].
C.-Y. Zhang et al., Dynamical charged black hole spontaneous scalarization in anti-de Sitter spacetimes, Phys. Rev. D 104 (2021) 084089 [arXiv:2103.13599] [INSPIRE].
C.-Y. Zhang et al., Dynamical transitions in scalarization and descalarization through black hole accretion, Phys. Rev. D 106 (2022) L061501 [arXiv:2204.09260] [INSPIRE].
Q. Chen et al., Descalarization by quenching charged hairy black hole in asymptotically AdS spacetime, JHEP 01 (2023) 062 [arXiv:2210.14539] [INSPIRE].
Q. Chen et al., Nonlinear dynamics of hot, cold, and bald Einstein-Maxwell-scalar black holes in AdS spacetime, Phys. Rev. D 108 (2023) 084016 [arXiv:2307.03060] [INSPIRE].
P. Hayden, O. Parrikar and J. Sorce, The Markov gap for geometric reflected entropy, JHEP 10 (2021) 047 [arXiv:2107.00009] [INSPIRE].
Y. Lu and J. Lin, The Markov gap in the presence of islands, JHEP 03 (2023) 043 [arXiv:2211.06886] [INSPIRE].
M. Afrasiar, J.K. Basak, A. Chandra and G. Sengupta, Reflected entropy for communicating black holes. Part I. Karch-Randall braneworlds, JHEP 02 (2023) 203 [arXiv:2211.13246] [INSPIRE].
M. Afrasiar, J.K. Basak, A. Chandra and G. Sengupta, Reflected entropy for communicating black holes II: Planck braneworlds, Eur. Phys. J. C 84 (2024) 250 [arXiv:2302.12810] [INSPIRE].
J.K. Basak et al., Holographic reflected entropy and islands in interface CFTs, JHEP 05 (2024) 143 [arXiv:2312.12512] [INSPIRE].
K. Kawabata et al., Dynamical quantum phase transitions in Sachdev-Ye-Kitaev Lindbladians, Phys. Rev. B 108 (2023) 075110 [arXiv:2210.04093] [INSPIRE].
A. Kulkarni, T. Numasawa and S. Ryu, Lindbladian dynamics of the Sachdev-Ye-Kitaev model, Phys. Rev. B 106 (2022) 075138 [arXiv:2112.13489] [INSPIRE].
A.M. García-García, L. Sá, J.J.M. Verbaarschot and J.P. Zheng, Keldysh wormholes and anomalous relaxation in the dissipative Sachdev-Ye-Kitaev model, Phys. Rev. D 107 (2023) 106006 [arXiv:2210.01695] [INSPIRE].
Acknowledgments
We are grateful to Pritam Chattopadhyay, Sayantan Choudhury, and Yuan Sun for helpful discussions. Yuxuan Liu is supported by the Beijing Natural Science Foundation under Grant No. 1222031 and the National Natural Science Foundation of China under Grant No. 12035016. He is particularly grateful to his wife Peiwen Cao. Shao-Kai Jian is supported by a startup fund at Tulane University. Yi Ling is supported in part by the Beijing Natural Science Foundation under Grant No. 1222031, and the Innovative Projects of Science and Technology at IHEP. He is also supported by the National Natural Science Foundation of China under Grant No. 12035016 and 12275275. Zhuo-Yu Xian is funded by DFG through the Collaborative Research Center SFB 1170 ToCoTronics, Project-ID 258499086–SFB 1170, as well as by Germany’s Excellence Strategy through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter — ct.qmat (EXC 2147, projectid 390858490). Zhuo-Yu Xian also acknowledges support from the National Natural Science Foundation of China under Grant No. 12075298.
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: 2401.04706
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., Jian, SK., Ling, Y. et al. Entanglement inside a black hole before the Page time. J. High Energ. Phys. 2024, 43 (2024). https://doi.org/10.1007/JHEP07(2024)043
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2024)043