Abstract
Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.
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, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [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].
X. Dong, The gravity dual of Renyi entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [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 and A. Lewkowycz, Entropy, extremality, Euclidean variations, and the equations of motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [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].
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].
C. Akers, N. Engelhardt and D. Harlow, Simple holographic models of black hole evaporation, JHEP 08 (2020) 032 [arXiv:1910.00972] [INSPIRE].
M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell and D. Wakeham, Information radiation in BCFT models of black holes, JHEP 05 (2020) 004 [arXiv:1910.12836] [INSPIRE].
H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, Information flow in black hole evaporation, JHEP 03 (2020) 152 [arXiv:1911.03402] [INSPIRE].
R. Bousso and M. Tomašević, Unitarity from a smooth horizon?, arXiv:1911.06305 [INSPIRE].
A. Almheiri, R. Mahajan and J.E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020) 001 [arXiv:1911.09666] [INSPIRE].
M. Alishahiha, A. Faraji Astaneh and A. Naseh, Island in the presence of higher derivative terms, arXiv:2005.08715 [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].
T. Anegawa and N. Iizuka, Notes on islands in asymptotically flat 2d dilaton black holes, JHEP 07 (2020) 036 [arXiv:2004.01601] [INSPIRE].
K. Hashimoto, N. Iizuka and Y. Matsuo, Islands in Schwarzschild black holes, JHEP 06 (2020) 085 [arXiv:2004.05863] [INSPIRE].
T. Hartman, E. Shaghoulian and A. Strominger, Islands in asymptotically flat 2D gravity, JHEP 07 (2020) 022 [arXiv:2004.13857] [INSPIRE].
S. Ghosh and S. Raju, Quantum information measures for restricted sets of observables, Phys. Rev. D 98 (2018) 046005 [arXiv:1712.09365] [INSPIRE].
A. Laddha, S.G. Prabhu, S. Raju and P. Shrivastava, The holographic nature of null infinity, arXiv:2002.02448 [INSPIRE].
C. Krishnan, V. Patil and J. Pereira, Page curve and the information paradox in flat space, arXiv:2005.02993 [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [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].
R. Jackiw, Lower dimensional gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [INSPIRE].
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
J. Cotler, C.-M. Jian, X.-L. Qi and F. Wilczek, Superdensity operators for spacetime quantum mechanics, JHEP 09 (2018) 093 [arXiv:1711.03119] [INSPIRE].
X. Dong and D. Marolf, One-loop universality of holographic codes, JHEP 03 (2020) 191 [arXiv:1910.06329] [INSPIRE].
S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, volume 1, Cambridge University Press, Cambridge, U.K. (1973).
D.N. Page, Density matrix of the universe, Phys. Rev. D 34 (1986) 2267 [INSPIRE].
S.W. Hawking, The density matrix of the universe, Phys. Scripta T 15 (1987) 151 [INSPIRE].
A.O. Bärvinsky, C. Deffayet and A. Kamenshchik, Density matrix of the universe reloaded: origin of inflation and cosmological acceleration, in 15th International Seminar on High Energy Physics, (2008) [arXiv:0810.5659] [INSPIRE].
J. Maldacena, G.J. Turiaci and Z. Yang, Two dimensional nearly de Sitter gravity, arXiv:1904.01911 [INSPIRE].
J. Maldacena, A simple model of quantum holography, talks at Strings, (2019).
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
J.B. Hartle and S.W. Hawking, Wave function of the universe, Phys. Rev. D 28 (1983) 2960 [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
T. Hertog and J. Hartle, Holographic no-boundary measure, JHEP 05 (2012) 095 [arXiv:1111.6090] [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
Z. Yang, The quantum gravity dynamics of near extremal black holes, JHEP 05 (2019) 205 [arXiv:1809.08647] [INSPIRE].
Y. Chen, V. Gorbenko and J. Maldacena, Bra-ket wormholes in gravitationally prepared states, arXiv:2007.16091 [INSPIRE].
J. Maldacena and A. Milekhin, SYK wormhole formation in real time, arXiv:1912.03276 [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
T. Faulkner, The entanglement Renyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
J. Polchinski, The black hole information problem, in New frontiers in fields and strings, World Scientific, Singapore (2016).
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
H.F. Jia and M. Rangamani, Petz reconstruction in random tensor networks, arXiv:2006.12601 [INSPIRE].
J. Cotler, X. Han, X.-L. Qi and Z. Yang, Quantum causal influence, JHEP 07 (2019) 042 [arXiv:1811.05485] [INSPIRE].
G.T. Horowitz and J.M. Maldacena, The black hole final state, JHEP 02 (2004) 008 [hep-th/0310281] [INSPIRE].
K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
T. Anous, J. Kruthoff and R. Mahajan, Density matrices in quantum gravity, arXiv:2006.17000 [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].
P. Hayden and G. Penington, Learning the alpha-bits of black holes, JHEP 12 (2019) 007 [arXiv:1807.06041] [INSPIRE].
C.-F. Chen, G. Penington and G. Salton, Entanglement wedge reconstruction using the Petz map, JHEP 01 (2020) 168 [arXiv:1902.02844] [INSPIRE].
Y. Chen, Pulling out the island with modular flow, JHEP 03 (2020) 033 [arXiv:1912.02210] [INSPIRE].
A. Gilyén, S. Lloyd, I. Marvian, Y. Quek and M.M. Wilde, Quantum algorithm for Petz recovery channels and pretty good measurements, arXiv:2006.16924 [INSPIRE].
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: 2007.02987
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
Dong, X., Qi, XL., Shangnan, Z. et al. Effective entropy of quantum fields coupled with gravity. J. High Energ. Phys. 2020, 52 (2020). https://doi.org/10.1007/JHEP10(2020)052
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2020)052