Abstract
We study the reflected entropy SR in the West Coast Model, a toy model of black hole evaporation consisting of JT gravity coupled to end-of-the-world branes. We demonstrate the validity of the holographic duality relating it to the entanglement wedge cross section away from phase transitions. Further, we analyze the important non-perturbative effects that smooth out the discontinuity in the SR phase transition. By performing the gravitational path integral, we obtain the reflected entanglement spectrum analytically. The spectrum takes a simple form consisting of superselection sectors, which we interpret as a direct sum of geometries, a disconnected one and a connected one involving a closed universe. We find that area fluctuations of O (\( \sqrt{G_N} \)) spread out the SR phase transition in the canonical ensemble, analogous to the entanglement entropy phase transition. We also consider a Renyi generalization of the reflected entropy and show that the location of the phase transition varies as a function of the Renyi parameter.
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.D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [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].
A. Almheiri, T. Hartman, J.M. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [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].
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].
C. Akers and G. Penington, Quantum minimal surfaces from quantum error correction, arXiv:2109.14618 [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178 [arXiv:1905.00577] [INSPIRE].
C. Akers and P. Rath, Entanglement Wedge Cross Sections Require Tripartite Entanglement, JHEP 04 (2020) 208 [arXiv:1911.07852] [INSPIRE].
V. Chandrasekaran, M. Miyaji and P. Rath, Including contributions from entanglement islands to the reflected entropy, Phys. Rev. D 102 (2020) 086009 [arXiv:2006.10754] [INSPIRE].
P. Hayden, O. Parrikar and J. Sorce, The Markov gap for geometric reflected entropy, JHEP 10 (2021) 047 [arXiv:2107.00009] [INSPIRE].
L. Vidmar and M. Rigol, Entanglement Entropy of Eigenstates of Quantum Chaotic Hamiltonians, Phys. Rev. Lett. 119 (2017) 220603 [arXiv:1708.08453] [INSPIRE].
C. Murthy and M. Srednicki, Structure of chaotic eigenstates and their entanglement entropy, Phys. Rev. E 100 (2019) 022131 [arXiv:1906.04295] [INSPIRE].
X. Dong and H. Wang, Enhanced corrections near holographic entanglement transitions: a chaotic case study, JHEP 11 (2020) 007 [arXiv:2006.10051] [INSPIRE].
D. Marolf, S. Wang and Z. Wang, Probing phase transitions of holographic entanglement entropy with fixed area states, JHEP 12 (2020) 084 [arXiv:2006.10089] [INSPIRE].
C. Akers, T. Faulkner, S. Lin and P. Rath, Reflected entropy in random tensor networks, arXiv:2112.09122 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
N. Engelhardt and A.C. Wall, Decoding the Apparent Horizon: Coarse-Grained Holographic Entropy, Phys. Rev. Lett. 121 (2018) 211301 [arXiv:1706.02038] [INSPIRE].
N. Engelhardt and A.C. Wall, Coarse Graining Holographic Black Holes, JHEP 05 (2019) 160 [arXiv:1806.01281] [INSPIRE].
N. Engelhardt and Å. Folkestad, Canonical purification of evaporating black holes, Phys. Rev. D 105 (2022) 086010 [arXiv:2201.08395] [INSPIRE].
C. Akers and G. Penington, Leading order corrections to the quantum extremal surface prescription, JHEP 04 (2021) 062 [arXiv:2008.03319] [INSPIRE].
H. Shapourian, S. Liu, J. Kudler-Flam and A. Vishwanath, Entanglement Negativity Spectrum of Random Mixed States: A Diagrammatic Approach, PRX Quantum 2 (2021) 030347 [arXiv:2011.01277] [INSPIRE].
X. Dong, S. McBride and W.W. Weng, Replica Wormholes and Holographic Entanglement Negativity, arXiv:2110.11947 [INSPIRE].
C. Akers and P. Rath, Holographic Renyi Entropy from Quantum Error Correction, JHEP 05 (2019) 052 [arXiv:1811.05171] [INSPIRE].
X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, JHEP 10 (2019) 240 [arXiv:1811.05382] [INSPIRE].
X. Dong and D. Marolf, One-loop universality of holographic codes, JHEP 03 (2020) 191 [arXiv:1910.06329] [INSPIRE].
A. Lewkowycz and J.M. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
X. Dong, The Gravity Dual of Renyi Entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [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].
C.-F. Chen, G. Penington and G. Salton, Entanglement Wedge Reconstruction using the Petz Map, JHEP 01 (2020) 168 [arXiv:1902.02844] [INSPIRE].
H. Liu and S. Vardhan, Entanglement entropies of equilibrated pure states in quantum many-body systems and gravity, PRX Quantum 2 (2021) 010344 [arXiv:2008.01089] [INSPIRE].
S. Vardhan, J. Kudler-Flam, H. Shapourian and H. Liu, Mixed-state entanglement and information recovery in thermalized states and evaporating black holes, arXiv:2112.00020 [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: 2201.11730
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
Akers, C., Faulkner, T., Lin, S. et al. The Page curve for reflected entropy. J. High Energ. Phys. 2022, 89 (2022). https://doi.org/10.1007/JHEP06(2022)089
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2022)089