Abstract
The Python’s Lunch conjecture for the complexity of bulk reconstruction involves two types of nonminimal quantum extremal surfaces (QESs): bulges and throats, which differ by their local properties. The conjecture relies on the connection between bulk spatial geometry and quantum codes: a constricting geometry from bulge to throat encodes the bulk state nonisometrically, and so requires an exponentially complex Grover search to decode. However, thus far, the Python’s Lunch conjecture is only defined for spacetimes where all QESs are spacelike-separated from one another. Here we explicitly construct (time-reflection symmetric) spacetimes featuring both timelike-separated bulges and timelike-separated throats. Interestingly, all our examples also feature a third type of QES, locally resembling a de Sitter bifurcation surface, which we name a bounce. By analyzing the Hessian of generalized entropy at a QES, we argue that this classification into throats, bulges and bounces is exhaustive. We then propose an updated Python’s Lunch conjecture that can accommodate general timelike-separated QESs and bounces. Notably, our proposal suggests that the gravitational analogue of a tensor network is not necessarily the time-reflection symmetric slice, even when one exists.
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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].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [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].
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].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP 07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
J. Cotler et al., Entanglement Wedge Reconstruction via Universal Recovery Channels, Phys. Rev. X 9 (2019) 031011 [arXiv:1704.05839] [INSPIRE].
D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
P. Hayden and G. Penington, Learning the Alpha-bits of Black Holes, JHEP 12 (2019) 007 [arXiv:1807.06041] [INSPIRE].
C. Akers, S. Leichenauer and A. Levine, Large Breakdowns of Entanglement Wedge Reconstruction, Phys. Rev. D 100 (2019) 126006 [arXiv:1908.03975] [INSPIRE].
C. Akers and G. Penington, Leading order corrections to the quantum extremal surface prescription, JHEP 04 (2021) 062 [arXiv:2008.03319] [INSPIRE].
C. Akers and G. Penington, Quantum minimal surfaces from quantum error correction, SciPost Phys. 12 (2022) 157 [arXiv:2109.14618] [INSPIRE].
C. Akers, A. Levine, G. Penington and E. Wildenhain, One-shot holography, arXiv:2307.13032 [INSPIRE].
N. Engelhardt, G. Penington and A. Shahbazi-Moghaddam, Finding pythons in unexpected places, Class. Quant. Grav. 39 (2022) 094002 [arXiv:2105.09316] [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].
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].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [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].
P. Hayden et al., Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
L.K. Grover, A fast quantum mechanical algorithm for database search, quant-ph/9605043 [INSPIRE].
V.E. Hubeny and H. Maxfield, Holographic probes of collapsing black holes, JHEP 03 (2014) 097 [arXiv:1312.6887] [INSPIRE].
R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [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].
A. Shahbazi-Moghaddam, Restricted Quantum Focusing, arXiv:2212.03881 [INSPIRE].
N. Engelhardt and A.C. Wall, Coarse Graining Holographic Black Holes, JHEP 05 (2019) 160 [arXiv:1806.01281] [INSPIRE].
N. Engelhardt and S. Fischetti, Surface Theory: the Classical, the Quantum, and the Holographic, Class. Quant. Grav. 36 (2019) 205002 [arXiv:1904.08423] [INSPIRE].
L. Andersson, M. Mars and W. Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys. 12 (2008) 853 [arXiv:0704.2889] [INSPIRE].
M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Springer New York (1999).
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. Marolf, A.C. Wall and Z. Wang, Restricted Maximin surfaces and HRT in generic black hole spacetimes, JHEP 05 (2019) 127 [arXiv:1901.03879] [INSPIRE].
C. Akers, N. Engelhardt, G. Penington and M. Usatyuk, Quantum Maximin Surfaces, JHEP 08 (2020) 140 [arXiv:1912.02799] [INSPIRE].
R. Bousso and A. Shahbazi-Moghaddam, Island Finder and Entropy Bound, Phys. Rev. D 103 (2021) 106005 [arXiv:2101.11648] [INSPIRE].
C.-F. Chen, G. Penington and G. Salton, Entanglement Wedge Reconstruction using the Petz Map, JHEP 01 (2020) 168 [arXiv:1902.02844] [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].
S. Fischetti, D. Marolf and A.C. Wall, A paucity of bulk entangling surfaces: AdS wormholes with de Sitter interiors, Class. Quant. Grav. 32 (2015) 065011 [arXiv:1409.6754] [INSPIRE].
A. Goel, H.T. Lam, G.J. Turiaci and H. Verlinde, Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK, JHEP 02 (2019) 156 [arXiv:1807.03916] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
L. Susskind, Three Lectures on Complexity and Black Holes, Springer (2018) [https://doi.org/10.1007/978-3-030-45109-7] [arXiv:1810.11563] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [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 et al., Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
Acknowledgments
It is a pleasure to thank A. Folkestad, D. Harlow, A. Levine, and Z. Yang for valuable discussions. NE is supported in part by NSF grant no. PHY-2011905, by the U.S. Department of Energy under Early Career Award DE-SC0021886, by the John Templeton Foundation via the Black Hole Initiative, by the Sloan Foundation, by the Heising-Simons Foundation, and by funds from the MIT physics department. GP was supported by the University of California, Berkeley; by the Department of Energy through DE-SC0019380 and DE-FOA-0002563; by AFOSR award FA9550-22-1-0098; and by an IBM Einstein Fellowship at the Institute for Advanced Study. ASM is supported by the National Science Foundation under Award Number 2014215.
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Engelhardt, N., Penington, G. & Shahbazi-Moghaddam, A. Twice upon a time: timelike-separated quantum extremal surfaces. J. High Energ. Phys. 2024, 33 (2024). https://doi.org/10.1007/JHEP01(2024)033
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DOI: https://doi.org/10.1007/JHEP01(2024)033