Abstract
When the bulk geometry in AdS/CFT contains a black hole, boundary subregions may be sufficient to reconstruct certain bulk operators if and only if the black hole microstate is known, an example of state dependence. Reconstructions exist for any microstate, but no reconstruction works for all microstates. We refine this dichotomy, demonstrating that the same boundary operator can often be used for large subspaces of black hole microstates, corresponding to a constant fraction α of the black hole entropy. In the Schrödinger picture, the boundary subregion encodes the α-bits (a concept from quantum information) of a bulk region containing the black hole and bounded by extremal surfaces. These results have important consequences for the structure of AdS/CFT and for quantum information. Firstly, they imply that the bulk reconstruction is necessarily only approximate and allow us to place non-perturbative lower bounds on the error when doing so. Second, they provide a simple and tractable limit in which the entanglement wedge is state dependent, but in a highly controlled way. Although the state dependence of operators comes from ordinary quantum error correction, there are clear connections to the Papadodimas-Raju proposal for understanding operators behind black hole horizons. In tensor network toy models of AdS/CFT, we see how state dependence arises from the bulk operator being ‘pushed’ through the black hole itself. Finally, we show that black holes provide the first ‘explicit’ examples of capacity-achieving α-bit codes. Unintuitively, Hawking radiation always reveals the α-bits of a black hole as soon as possible. In an appendix, we apply a result from the quantum information literature to prove that entanglement wedge reconstruction can be made exact to all orders in 1/N.
Article PDF
Similar content being viewed by others
References
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys.354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
T. Hirata and T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy, JHEP02 (2007) 042 [hep-th/0608213] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav.42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
E. Verlinde and H. Verlinde, Black Hole Entanglement and Quantum Error Correction, JHEP10 (2013) 107 [arXiv:1211.6913] [INSPIRE].
S. Lloyd and J. Preskill, Unitarity of black hole evaporation in final-state projection models, JHEP08 (2014) 126 [arXiv:1308.4209] [INSPIRE].
B. Yoshida and A. Kitaev, Efficient decoding for the Hayden-Preskill protocol, arXiv:1710.03363 [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP06 (2015) 149 [arXiv:1503.06237] [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].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP10 (2013) 212 [arXiv:1211.6767] [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].
P. Hayden and G. Penington, Approximate quantum error correction revisited: Introducing the alpha-bit, arXiv:1706.09434.
N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
C. Bény, Conditions for the approximate correction of algebras, in proceedings of the Theory of quantum computation, communication and cryptography: 4th workshop, TQC 2009, Waterloo, Canada, 11–13 May 2009, Springer, Lect. Notes Comput. Sci.5906 (2009) 66.
A. Winter, Quantum and classical message identification via quantum channels, in Festschrift “A.S. Holevo 60”, O. Hirota ed., Rinton Press (2004), pp. 171–188, reprinted in Quant. Inf. Comput.4 (2004) 563 [quant-ph/0401060].
P. Hayden and A. Winter, Weak decoupling duality and quantum identification, IEEE Trans. Inf. Theory58 (2012) 4914.
D. Kretschmann and R.F. Werner, Tema con variazioni: quantum channel capacity, New J. Phys.6 (2004) 26.
D. Kretschmann, D. Schlingemann and R.F. Werner, The information-disturbance tradeoff and the continuity of Stinespring’s representation, quant-ph/0605009.
D.N. Page, Time Dependence of Hawking Radiation Entropy, JCAP09 (2013) 028 [arXiv:1301.4995] [INSPIRE].
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations and the Equations of Motion, JHEP01 (2018) 081 [arXiv:1705.08453] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [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].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
A.C. Wall, Maximin Surfaces and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav.31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle and M. Walter, Entanglement Wedge Reconstruction via Universal Recovery Channels, Phys. Rev.X 9 (2019) 031011 [arXiv:1704.05839] [INSPIRE].
F. Hiai, M. Ohya and M. Tsukada, Sufficiency, KMS condition and relative entropy in von Neumann algebras, in Selected Papers of M. Ohya , World Scientific (2008), pp. 420–430.
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
N. Bao and H. Ooguri, Distinguishability of black hole microstates, Phys. Rev.D 96 (2017) 066017 [arXiv:1705.07943] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev.D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Tensor Networks from Kinematic Space, JHEP07 (2016) 100 [arXiv:1512.01548] [INSPIRE].
E.P. Verlinde, Emergent Gravity and the Dark Universe, SciPost Phys.2 (2017) 016 [arXiv:1611.02269] [INSPIRE].
T.J. Osborne and D.E. Stiegemann, Dynamics for holographic codes, arXiv:1706.08823 [INSPIRE].
S. Popescu, A.J. Short and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys.2 (2006) 754.
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev.A 43 (1991) 2046.
M. Srednicki, Chaos and quantum thermalization, Phys. Rev.E 50 (1994) 888.
D. Marolf and J. Polchinski, Gauge/Gravity Duality and the Black Hole Interior, Phys. Rev. Lett.111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].
R. Haag, N.M. Hugenholtz and M. Winnink, On the Equilibrium states in quantum statistical mechanics, Commun. Math. Phys.5 (1967) 215 [INSPIRE].
R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jap.12 (1957) 570 [INSPIRE].
W.R. Kelly, Bulk Locality and Entanglement Swapping in AdS/CFT, JHEP03 (2017) 153 [arXiv:1610.00669] [INSPIRE].
H. Reeh and S. Schlieder, Bemerkungen zur Unit¨ar¨aquivalenz von Lorentzinvarianten Feldern, Nuovo Cim.22 (1961) 1051 [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys.61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
D. Harlow, Aspects of the Papadodimas-Raju Proposal for the Black Hole Interior, JHEP11 (2014) 055 [arXiv:1405.1995] [INSPIRE].
I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS 2gravity, arXiv:1707.02325 [INSPIRE].
J. de Boer, R. van Breukelen, S.F. Lokhande, K. Papadodimas and E. Verlinde, On the interior geometry of a typical black hole microstate, JHEP05 (2019) 010 [arXiv:1804.10580] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
M. Junge, R. Renner, D. Sutter, M.M. Wilde and A. Winter, Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy, Ann. Henri Poincaŕe19 (2018) 2955 [arXiv:1509.07127] [INSPIRE].
O. Fawzi, P. Hayden and P. Sen, From low-distortion norm embeddings to explicit uncertainty relations and efficient information locking, J. ACM60 (2013) 44.
C. Bény, A. Kempf and D.W. Kribs, Generalization of quantum error correction via the Heisenberg picture, Phys. Rev. Lett.98 (2007) 100502.
C. Bény, Z. Zimborás and F. Pastawski, Approximate recovery with locality and symmetry constraints, arXiv:1806.10324 [INSPIRE].
C. Bény and O. Oreshkov, General conditions for approximate quantum error correction and near-optimal recovery channels, Phys. Rev. Lett.104 (2010) 120501.
J. Tyson, Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates, J. Math. Phys.51 (2010) 092204 [arXiv:0907.3386].
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1807.06041
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Hayden, P., Penington, G. Learning the Alpha-bits of black holes. J. High Energ. Phys. 2019, 7 (2019). https://doi.org/10.1007/JHEP12(2019)007
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2019)007