Abstract
Recent work has established a route towards the semiclassical validity of the Page curve, and so provided evidence that information escapes an evaporating black hole. However, a protocol to explicitly recover and make practical use of that information in the classical limit has not yet been given. In this paper, we describe such a protocol, showing that an observer may reconstruct the phase space of the black hole interior by measuring the Uhlmann phase of the Hawking radiation. The process of black hole formation and evaporation provides an invertible map between this phase space and the space of initial matter configurations. Thus, all classical information is explicitly recovered. We assume in this paper that replica wormholes contribute to the gravitational path integral.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
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, Time Dependence of Hawking Radiation Entropy, JCAP 09 (2013) 028 [arXiv:1301.4995] [INSPIRE].
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].
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, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [arXiv:2006.06872] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [INSPIRE].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [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].
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].
J. Camps, Generalized entropy and higher derivative Gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].
X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [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].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [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].
A. Belin, A. Lewkowycz and G. Sárosi, The boundary dual of the bulk symplectic form, Phys. Lett. B 789 (2019) 71 [arXiv:1806.10144] [INSPIRE].
A. Belin, A. Lewkowycz and G. Sárosi, Complexity and the bulk volume, a new York time story, JHEP 03 (2019) 044 [arXiv:1811.03097] [INSPIRE].
J. Kirklin, The Holographic Dual of the Entanglement Wedge Symplectic Form, JHEP 01 (2020) 071 [arXiv:1910.00457] [INSPIRE].
A. Uhlmann, The Metric of Bures and the Geometric Phase, in Groups and Related Topics: Proceedings of the First Max Born Symposium, R. Gielerak, J. Lukierski and Z. Popowicz eds., Springer, Dordrecht The Netherlands (1992), pg. 267 [ISBN:978-94-011-2801-8].
A. Uhlmann, Gauge field governing parallel transport along mixed states, Lett. Math. Phys. 21 (1991) 229 [INSPIRE].
A. Uhlmann, The “transition probability” in the state space of a*-algebra, Rep. Math. Phys. 9 (1976) 273.
S.L. Braunstein and C.M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72 (1994) 3439 [INSPIRE].
R. Jozsa, Fidelity for Mixed Quantum States, J. Mod. Opt. 41 (1994) 2315.
D. Bures, An Extension of Kakutani’s Theorem on Infinite Product Measures to the Tensor Product of Semifinite w*-Algebras, Trans. Am. Math. Soc. 135 (1969) 199.
C.W. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Phys. Lett. A 25 (1967) 101.
S.B. Giddings and G.J. Turiaci, Wormhole calculus, replicas, and entropies, JHEP 09 (2020) 194 [arXiv:2004.02900] [INSPIRE].
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 [arXiv:2002.08950] [INSPIRE].
D. Marolf and H. Maxfield, Observations of Hawking radiation: the Page curve and baby universes, JHEP 04 (2021) 272 [arXiv:2010.06602] [INSPIRE].
J. Yngvason, The Role of type-III factors in quantum field theory, Rept. Math. Phys. 55 (2005) 135 [math-ph/0411058] [INSPIRE].
C.G. Callan, Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
R.E. Peierls, The Commutation laws of relativistic field theory, Proc. Roy. Soc. Lond. A 214 (1952) 143 [INSPIRE].
B.S. DeWitt, The spacetime approach to quantum field theory, in Relativity, groups and topology: Proceedings of 40th Summer School of Theoretical Physics - Session 40. Vol. 2, Les Houches France (1983), pg. 381.
B.S. DeWitt, The global approach to quantum field theory. Vol. 1, 2, Int. Ser. Monogr. Phys. 114 (2003) 1.
P.G. Bergmann and R. Schiller, Classical and Quantum Field Theories in the Lagrangian Formalism, Phys. Rev. 89 (1953) 4 [INSPIRE].
C. Crnković and E. Witten, Covariant description of canonical formalism in geometrical theories, in: Three Hundred Years of Gravitation, S.W. Hawking and W. Israel eds., Cambridge University Press, Cambridge U.K. (1989), pg. 676.
C. Crnković, Symplectic Geometry of the Covariant Phase Space, Superstrings and Superspace, Class. Quant. Grav. 5 (1988) 1557 [INSPIRE].
G.J. Zuckerman, Action principles and global geometry, Conf. Proc. C 8607214 (1986) 259 [INSPIRE].
A. Ashtekar and A. Magnon-Ashtekar, On the symplectic structure of general relativity, Commun. Math. Phys. 86 (1982) 55.
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
J.D. Brown and J.W. York, Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].
D.M. Marolf, Poisson brackets on the space of histories, Annals Phys. 236 (1994) 374 [hep-th/9308141] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
R.M. Wald and A. Zoupas, A General definition of ’conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
S. Hollands and D. Marolf, Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry, Class. Quant. Grav. 24 (2007) 2301 [gr-qc/0611044] [INSPIRE].
D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146 [arXiv:1906.08616] [INSPIRE].
D. Harlow and P. Hayden, Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
J. Åberg, D. Kult, E. Sjöqvist and D.K.L. Oi, Operational approach to the Uhlmann holonomy, Phys. Rev. A 75 (2007) 032106 [quant-ph/0608185].
O. Viyuela, A. Rivas, S. Gasparinetti, A. Wallraff, S. Filipp and M.A. Martin-Delgado, Observation of topological Uhlmann phases with superconducting qubits, arXiv:1607.08778.
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: 2011.07086
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
Kirklin, J. Islands and Uhlmann phase: explicit recovery of classical information from evaporating black holes. J. High Energ. Phys. 2022, 119 (2022). https://doi.org/10.1007/JHEP01(2022)119
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2022)119