Abstract
We revisit information retrieval from evaporating black holes in the Hayden-Preskill protocol, treating the black hole dynamics as Haar-random. We compute, down to the first exponentially suppressed terms, all integer-indexed Rényi mutual informations between a black hole, its radiation, and a reference that catalogues Alice’s diaries. We find that dropping a diary into a young black hole effectively delays the Page time. We also compute the radiation: diary reflected Rényi entropies, and identify a technical reason why they cannot be continued to the reflected entropy by the replica trick.
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P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].
J.D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974) 3292 [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [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. 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].
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
D. Harlow, The Ryu-Takayanagi formula from quantum error correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
B. Yoshida and A. Kitaev, Efficient decoding for the Hayden-Preskill protocol, arXiv:1710.03363 [INSPIRE].
B. Yoshida, Soft mode and interior operator in the Hayden-Preskill thought experiment, Phys. Rev. D 100 (2019) 086001 [arXiv:1812.07353] [INSPIRE].
B. Yoshida, Recovery algorithms for Clifford Hayden-Preskill problem, arXiv:2106.15628 [INSPIRE].
J.-M. Bae, S. Kang, D.-H. Yeom and H. Zoe, Demonstration of the Hayden-Preskill protocol via mutual information, J. Korean Phys. Soc. 75 (2019) 941 [arXiv:1907.13290] [INSPIRE].
S.H. Lie, Y.S. Teo and H. Jeong, Repeated extraction of scrambled quantum data: sustainability of the Hayden-Preskill type protocols, arXiv:2204.00374 [INSPIRE].
H. Tajima and K. Saito, Universal limitation of quantum information recovery: symmetry versus coherence, arXiv:2103.01876 [INSPIRE].
N. Bao and Y. Kikuchi, Hayden-Preskill decoding from noisy Hawking radiation, JHEP 02 (2021) 017 [arXiv:2009.13493] [INSPIRE].
Y. Nakata, E. Wakakuwa and M. Koashi, Black holes as clouded mirrors: the Hayden-Preskill protocol with symmetry, Quantum 7 (2023) 928 [arXiv:2007.00895] [INSPIRE].
R.J. Garcia, K. Bu and A. Jaffe, Resource theory of quantum scrambling, Proc. Nat. Acad. Sci. 120 (2023) e2217031120 [arXiv:2208.10477] [INSPIRE].
L. Leone et al., Retrieving information from a black hole using quantum machine learning, Phys. Rev. A 106 (2022) 062434 [arXiv:2206.06385] [INSPIRE].
S. Vardhan, J. Kudler-Flam, H. Shapourian and H. Liu, Mixed-state entanglement and information recovery in thermalized states and evaporating black holes, JHEP 01 (2023) 064 [arXiv:2112.00020] [INSPIRE].
Y. Nakata and M. Tezuka, Hayden-Preskill recovery in Hamiltonian systems, arXiv:2303.02010 [INSPIRE].
T. Hayata, Y. Hidaka and Y. Kikuchi, Diagnosis of information scrambling from Hamiltonian evolution under decoherence, Phys. Rev. D 104 (2021) 074518 [arXiv:2103.05179] [INSPIRE].
M.S. Blok et al., Quantum information scrambling on a superconducting Qutrit processor, Phys. Rev. X 11 (2021) 021010 [arXiv:2003.03307] [INSPIRE].
Y. Cheng et al., Realizing the Hayden-Preskill protocol with coupled Dicke models, Phys. Rev. Res. 2 (2020) 043024 [arXiv:1909.12568] [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178 [arXiv:1905.00577] [INSPIRE].
L. Piroli, C. Sünderhauf and X.-L. Qi, A random unitary circuit model for black hole evaporation, JHEP 04 (2020) 063 [arXiv:2002.09236] [INSPIRE].
D. Stanford, Z. Yang and S. Yao, Subleading Weingartens, JHEP 02 (2022) 200 [arXiv:2107.10252] [INSPIRE].
C. Akers et al., The black hole interior from non-isometric codes and complexity, arXiv:2207.06536 [INSPIRE].
M.P.A. Fisher, V. Khemani, A. Nahum and S. Vijay, Random quantum circuits, Ann. Rev. Condensed Matter Phys. 14 (2023) 335 [arXiv:2207.14280] [INSPIRE].
C. Akers, T. Faulkner, S. Lin and P. Rath, Reflected entropy in random tensor networks, JHEP 05 (2022) 162 [arXiv:2112.09122] [INSPIRE].
C. Akers and P. Rath, Entanglement wedge cross sections require tripartite entanglement, JHEP 04 (2020) 208 [arXiv:1911.07852] [INSPIRE].
P. Hayden, O. Parrikar and J. Sorce, The Markov gap for geometric reflected entropy, JHEP 10 (2021) 047 [arXiv:2107.00009] [INSPIRE].
Y. Zou et al., Universal tripartite entanglement in one-dimensional many-body systems, Phys. Rev. Lett. 126 (2021) 120501 [arXiv:2011.11864] [INSPIRE].
P. Hayden, M. Lemm and J. Sorce, Reflected entropy: not a correlation measure, Phys. Rev. A 107 (2023) L050401 [arXiv:2302.10208] [INSPIRE].
Y. Kusuki and K. Tamaoka, Entanglement wedge cross section from CFT: dynamics of local operator quench, JHEP 02 (2020) 017 [arXiv:1909.06790] [INSPIRE].
J. Kudler-Flam, Y. Kusuki and S. Ryu, Correlation measures and the entanglement wedge cross-section after quantum quenches in two-dimensional conformal field theories, JHEP 04 (2020) 074 [arXiv:2001.05501] [INSPIRE].
M. Moosa, Time dependence of reflected entropy in rational and holographic conformal field theories, JHEP 05 (2020) 082 [arXiv:2001.05969] [INSPIRE].
P. Bueno and H. Casini, Reflected entropy, symmetries and free fermions, JHEP 05 (2020) 103 [arXiv:2003.09546] [INSPIRE].
C. Berthiere, H. Chen, Y. Liu and B. Chen, Topological reflected entropy in Chern-Simons theories, Phys. Rev. B 103 (2021) 035149 [arXiv:2008.07950] [INSPIRE].
J. Kudler-Flam, Y. Kusuki and S. Ryu, The quasi-particle picture and its breakdown after local quenches: mutual information, negativity, and reflected entropy, JHEP 03 (2021) 146 [arXiv:2008.11266] [INSPIRE].
P. Bueno and H. Casini, Reflected entropy for free scalars, JHEP 11 (2020) 148 [arXiv:2008.11373] [INSPIRE].
H.A. Camargo et al., Long distance entanglement of purification and reflected entropy in conformal field theory, Phys. Rev. Lett. 127 (2021) 141604 [arXiv:2102.00013] [INSPIRE].
S. Dutta, T. Faulkner and S. Lin, The reflected entanglement spectrum for free fermions, JHEP 02 (2023) 223 [arXiv:2211.17255] [INSPIRE].
N. Bao and N. Cheng, Multipartite reflected entropy, JHEP 10 (2019) 102 [arXiv:1909.03154] [INSPIRE].
J. Chu, R. Qi and Y. Zhou, Generalizations of reflected entropy and the holographic dual, JHEP 03 (2020) 151 [arXiv:1909.10456] [INSPIRE].
N. Bao, A. Chatwin-Davies and G.N. Remmen, Entanglement wedge cross section inequalities from replicated geometries, JHEP 07 (2021) 113 [arXiv:2106.02640] [INSPIRE].
C. Akers, T. Faulkner, S. Lin and P. Rath, Reflected entropy in random tensor networks. Part II. A topological index from canonical purification, JHEP 01 (2023) 067 [arXiv:2210.15006] [INSPIRE].
D. Basu, H. Parihar, V. Raj and G. Sengupta, Entanglement negativity, reflected entropy, and anomalous gravitation, Phys. Rev. D 105 (2022) 086013 [Erratum ibid. 105 (2022) 129902] [arXiv:2202.00683] [INSPIRE].
P. Hayden et al., Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
P. Vemuri et al., Association of lifetime intellectual enrichment with cognitive decline in the older population, JAMA Neurol. 71 (2014) 1017.
B. Czech and S. Shuai, Holographic cone of average entropies, Commun. Phys. 5 (2022) 244 [arXiv:2112.00763] [INSPIRE].
V. Balasubramanian, A. Kar, C. Li and O. Parrikar, Quantum error correction in the black hole interior, JHEP 07 (2023) 189 [arXiv:2203.01961] [INSPIRE].
D. Weingarten, Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys. 19 (1978) 999 [INSPIRE].
G. Köstenberger, Weingarten calculus, arXiv:2101.00921.
R.P. Stanley, Enumerative combinatorics, Springer, New York, NY, U.S.A. (2012) [https://doi.org/10.1007/978-1-4615-9763-6].
P. Zinn-Justin, Jucys-Murphy elements and Weingarten matrices, Lett. Math. Phys. 91 (2009) 119 [arXiv:0907.2719].
Acknowledgments
An early version of the material in this paper was first presented by HT as a final project in class Quantum Information and Quantum Gravity, which BC taught (and SS attended) at Tsinghua University in Fall 2022. We thank other students in the class, especially Dachen Zhang, for useful discussions. BC and SS thank the organizers of the workshop ‘Quantum Information, Quantum Matter and Quantum Gravity’ (YITP-T-23-01) held at YITP, Kyoto University (2023), where this work was completed. The work of BC and SS is supported by an NSFC grant number 12042505, a BJNSF grant under the Gao Cengci Rencai Zizhu program, and a Dushi Zhuanxiang Fellowship.
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Czech, B., Shuai, S. & Tang, H. Entropies and reflected entropies in the Hayden-Preskill protocol. J. High Energ. Phys. 2024, 40 (2024). https://doi.org/10.1007/JHEP02(2024)040
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DOI: https://doi.org/10.1007/JHEP02(2024)040