Abstract
We study the quantum error correction properties of the black hole interior in a toy model for an evaporating black hole: Jackiw-Teitelboim gravity entangled with a non-gravitational bath. After the Page time, the black hole interior degrees of freedom in this system are encoded in the bath Hilbert space. We use the gravitational path integral to show that the interior density matrix is correctable against the action of quantum operations on the bath which (i) do not have prior access to details of the black hole microstates, and (ii) do not have a large, negative coherent information with respect to the maximally mixed state on the bath, with the lower bound controlled by the black hole entropy and code subspace dimension. Thus, the encoding of the black hole interior in the radiation is robust against generic, low-rank quantum operations. For erasure errors, gravity comes within an O(1) distance of saturating the Singleton bound on the tolerance of error correcting codes. For typical errors in the bath to corrupt the interior, they must have a rank that is a large multiple of the bath Hilbert space dimension, with the precise coefficient set by the black hole entropy and code subspace dimension.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
V. Balasubramanian, P. Kraus and A.E. Lawrence, Bulk versus boundary dynamics in anti-de Sitter space-time, Phys. Rev. D 59 (1999) 046003 [hep-th/9805171] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
V. Balasubramanian, S.B. Giddings and A.E. Lawrence, What do CFTs tell us about Anti-de Sitter space-times?, JHEP 03 (1999) 001 [hep-th/9902052] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
E. Verlinde and H. Verlinde, Black Hole Entanglement and Quantum Error Correction, JHEP 10 (2013) 107 [arXiv:1211.6913] [INSPIRE].
K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
D. Harlow, TASI Lectures on the Emergence of Bulk Physics in AdS/CFT, PoS TASI2017 (2018) 002 [arXiv:1802.01040] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [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].
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].
P. Hayden and G. Penington, Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit, Commun. Math. Phys. 374 (2020) 369 [arXiv:1706.09434] [INSPIRE].
P. Hayden and G. Penington, Learning the Alpha-bits of Black Holes, JHEP 12 (2019) 007 [arXiv:1807.06041] [INSPIRE].
C. Akers and G. Penington, Leading order corrections to the quantum extremal surface prescription, JHEP 04 (2021) 062 [arXiv:2008.03319] [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].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri et al., Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [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].
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].
V. Balasubramanian, A. Kar and T. Ugajin, Entanglement between two disjoint universes, JHEP 02 (2021) 136 [arXiv:2008.05274] [INSPIRE].
H. Geng et al., Information Transfer with a Gravitating Bath, SciPost Phys. 10 (2021) 103 [arXiv:2012.04671] [INSPIRE].
P. Gao, D.L. Jafferis and D.K. Kolchmeyer, An effective matrix model for dynamical end of the world branes in Jackiw-Teitelboim gravity, JHEP 01 (2022) 038 [arXiv:2104.01184] [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.R. Brown, H. Gharibyan, G. Penington and L. Susskind, The Python’s Lunch: geometric obstructions to decoding Hawking radiation, JHEP 08 (2020) 121 [arXiv:1912.00228] [INSPIRE].
I. Kim, E. Tang and J. Preskill, The ghost in the radiation: Robust encodings of the black hole interior, JHEP 06 (2020) 031 [arXiv:2003.05451] [INSPIRE].
N. Engelhardt, G. Penington and A. Shahbazi-Moghaddam, A world without pythons would be so simple, Class. Quant. Grav. 38 (2021) 234001 [arXiv:2102.07774] [INSPIRE].
N. Engelhardt, G. Penington and A. Shahbazi-Moghaddam, Finding pythons in unexpected places, Class. Quant. Grav. 39 (2022) 094002 [arXiv:2105.09316] [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th ed., Cambridge University Press, U.S.A. (2011).
B. Schumacher and M.A. Nielsen, Quantum data processing and error correction, Phys. Rev. A 54 (1996) 2629 [quant-ph/9604022] [INSPIRE].
J. Preskill, Quantum Shannon Theory, arXiv:1604.07450 [INSPIRE].
B. Schumacher and M.D. Westmoreland, Approximate quantum error correction, quant-ph/0112106 [https://doi.org/10.48550/arXiv.quant-ph/0112106].
D.W. Kribs, R. Laflamme, D. Poulin and M. Lesosky, Operator quantum error correction, quant-ph/0504189 [https://doi.org/10.48550/arXiv.quant-ph/0504189].
M.A. Nielsen and D. Poulin, Algebraic and information-theoretic conditions for operator quantum error correction, Phys. Rev. A 75 (2007) 064304.
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
D. Marolf et al., From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories, JHEP 06 (2018) 077 [arXiv:1709.10101] [INSPIRE].
X.-L. Qi, Entanglement island, miracle operators and the firewall, JHEP 01 (2022) 085 [arXiv:2105.06579] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
C. Akers and G. Penington, Quantum minimal surfaces from quantum error correction, SciPost Phys. 12 (2022) 157 [arXiv:2109.14618] [INSPIRE].
V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, The Library of Babel: On the origin of gravitational thermodynamics, JHEP 12 (2005) 006 [hep-th/0508023] [INSPIRE].
V. Balasubramanian, A. Kar, C. Li and O. Parrikar, Quantum error correction in the Python’s lunch, to appear.
K. Langhoff, C. Murdia and Y. Nomura, Multiverse in an inverted island, Phys. Rev. D 104 (2021) 086007 [arXiv:2106.05271] [INSPIRE].
R. Bousso and E. Wildenhain, Islands in closed and open universes, Phys. Rev. D 105 (2022) 086012 [arXiv:2202.05278] [INSPIRE].
E. Shaghoulian and L. Susskind, Entanglement in De Sitter space, JHEP 08 (2022) 198 [arXiv:2201.03603] [INSPIRE].
L. Anderson, O. Parrikar and R.M. Soni, Islands with gravitating baths: towards ER = EPR, JHEP 21 (2020) 226 [arXiv:2103.14746] [INSPIRE].
V. Balasubramanian, A. Kar and T. Ugajin, Entanglement between two gravitating universes, Class. Quant. Grav. 39 (2022) 174001 [arXiv:2104.13383] [INSPIRE].
Y. Zhao, Petz map and Python’s lunch, JHEP 11 (2020) 038 [arXiv:2003.03406] [INSPIRE].
A. Gilyén et al., Quantum Algorithm for Petz Recovery Channels and Pretty Good Measurements, Phys. Rev. Lett. 128 (2022) 220502 [arXiv:2006.16924] [INSPIRE].
H. Geng et al., Inconsistency of islands in theories with long-range gravity, JHEP 01 (2022) 182 [arXiv:2107.03390] [INSPIRE].
D. Bak, C. Kim, S.-H. Yi and J. Yoon, Python’s lunches in Jackiw-Teitelboim gravity with matter, JHEP 04 (2022) 175 [arXiv:2112.04224] [INSPIRE].
C. Bény, A. Kempf and D.W. Kribs, Quantum error correction of observables, Phys. Rev. A 76 (2007) 042303 [arXiv:0705.1574].
V. Balasubramanian et al., Complexity growth in integrable and chaotic models, JHEP 07 (2021) 011 [arXiv:2101.02209] [INSPIRE].
Acknowledgments
We thank Abhijit Gadde, Lampros Lamprou, Raghu Mahajan, Gautam Mandal, Alexey Milekhin, Shiraz Minwalla, Daniel Ranard, Pratik Rath, Pushkal Shrivastava, Douglas Stanford, Sandip Trivedi and Zhenbin Yang for useful discussions. We thank Abhijit Gadde for comments on a previous version of the paper. VB is supported in part by the Department of Energy through grant DE-SC0013528 and grant QuantISED DE-SC0020360, as well as the Simons Foundation through the It From Qubit Collaboration (Grant No. 38559). AK is supported by the Simons Foundation through the It from Qubit Collaboration. CL is supported by the Department of Energy through grants DE-SC0013528 and DE-SC0020360. OP is supported by the Department of Atomic Energy, Government of India, under project identification number RTI 4002.
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: 2203.01961
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
Balasubramanian, V., Kar, A., Li, C. et al. Quantum error correction in the black hole interior. J. High Energ. Phys. 2023, 189 (2023). https://doi.org/10.1007/JHEP07(2023)189
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2023)189