Abstract
Quantum error correction has given us a natural language for the emergence of spacetime, but the black hole interior poses a challenge for this framework: at late times the apparent number of interior degrees of freedom in effective field theory can vastly exceed the true number of fundamental degrees of freedom, so there can be no isometric (i.e. inner-product preserving) encoding of the former into the latter. In this paper we explain how quantum error correction nonetheless can be used to explain the emergence of the black hole interior, via the idea of “non-isometric codes protected by computational complexity”. We show that many previous ideas, such as the existence of a large number of “null states”, a breakdown of effective field theory for operations of exponential complexity, the quantum extremal surface calculation of the Page curve, post-selection, “state-dependent/state-specific” operator reconstruction, and the “simple entropy” approach to complexity coarse-graining, all fit naturally into this framework, and we illustrate all of these phenomena simultaneously in a soluble model.
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References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976) 2460 [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].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [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].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].
R. Bousso et al., Null Geodesics, Local CFT Operators and AdS/CFT for Subregions, Phys. Rev. D 88 (2013) 064057 [arXiv:1209.4641] [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].
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].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [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].
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].
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].
D. Harlow, The Ryu–Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
P. Hayden et al., Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
J. Cotler et al., Entanglement Wedge Reconstruction via Universal Recovery Channels, Phys. Rev. X 9 (2019) 031011 [arXiv:1704.05839] [INSPIRE].
P. Hayden and G. Penington, Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit, Commun. Math. Phys. 374 (2020) 369 [arXiv:1706.09434] [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].
S.D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
D. Marolf and A.C. Wall, Eternal Black Holes and Superselection in AdS/CFT, Class. Quant. Grav. 30 (2013) 025001 [arXiv:1210.3590] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
E. Verlinde and H. Verlinde, Black Hole Entanglement and Quantum Error Correction, JHEP 10 (2013) 107 [arXiv:1211.6913] [INSPIRE].
A. Almheiri et al., An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
D. Marolf and J. Polchinski, Gauge/Gravity Duality and the Black Hole Interior, Phys. Rev. Lett. 111 (2013) 171301 [arXiv:1307.4706] [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].
D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information, Rev. Mod. Phys. 88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].
J.V. Rocha, Evaporation of large black holes in AdS: Coupling to the evaporon, JHEP 08 (2008) 075 [arXiv:0804.0055] [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].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
Y. Kiem, H.L. Verlinde and E.P. Verlinde, Black hole horizons and complementarity, Phys. Rev. D 52 (1995) 7053 [hep-th/9502074] [INSPIRE].
D.L. Jafferis, Bulk reconstruction and the Hartle-Hawking wavefunction, arXiv:1703.01519 [INSPIRE].
P. Hayden and G. Penington, Learning the Alpha-bits of Black Holes, JHEP 12 (2019) 007 [arXiv:1807.06041] [INSPIRE].
A. Almheiri, Holographic Quantum Error Correction and the Projected Black Hole Interior, arXiv:1810.02055 [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].
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. Harlow and P. Hayden, Quantum Computation vs. Firewalls, JHEP 06 (2013) 085 [arXiv:1301.4504] [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
L. Susskind, The Typical-State Paradox: Diagnosing Horizons with Complexity, Fortsch. Phys. 64 (2016) 84 [arXiv:1507.02287] [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].
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].
A. Bouland, B. Fefferman and U. Vazirani, Computational pseudorandomness, the wormhole growth paradox, and constraints on the AdS/CFT duality, arXiv:1910.14646 [INSPIRE].
I.H. Kim, E. Tang and J. Preskill, The ghost in the radiation: robust encodings of the black hole interior (invited paper), JHEP 06 (2020) 031 [arXiv:2003.05451] [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].
N. Engelhardt, G. Penington and A. Shahbazi-Moghaddam, Finding pythons in unexpected places, Class. Quant. Grav. 39 (2022) 094002 [arXiv:2105.09316] [INSPIRE].
A. Almheiri et al., Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
G.T. Horowitz and J.M. Maldacena, The black hole final state, JHEP 02 (2004) 008 [hep-th/0310281] [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].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
E.S. Meckes, The random matrix theory of the classical compact groups, vol. 218, Cambridge University Press (2019).
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
D. Harlow and D. Jafferis, The Factorization Problem in Jackiw-Teitelboim Gravity, JHEP 02 (2020) 177 [arXiv:1804.01081] [INSPIRE].
L. Susskind, Black Holes at Exp-time, arXiv:2006.01280 [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].
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].
K. Papadodimas and S. Raju, Remarks on the necessity and implications of state-dependence in the black hole interior, Phys. Rev. D 93 (2016) 084049 [arXiv:1503.08825] [INSPIRE].
J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018) [https://doi.org/10.1017/9781316848142] [INSPIRE].
E.P. Wigner, Remarks on the mind-body question, in Philosophical reflections and syntheses, Springer (1995) pp. 247–260.
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
N. Lashkari et al., Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644 [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
G. ’t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
V.E. Hubeny and M. Rangamani, Causal Holographic Information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].
W.R. Kelly and A.C. Wall, Coarse-grained entropy and causal holographic information in AdS/CFT, JHEP 03 (2014) 118 [arXiv:1309.3610] [INSPIRE].
N. Engelhardt and A.C. Wall, No Simple Dual to the Causal Holographic Information?, JHEP 04 (2017) 134 [arXiv:1702.01748] [INSPIRE].
E.T. Jaynes, Information Theory and Statistical Mechanics, Phys. Rev. 106 (1957) 620 [INSPIRE].
E.T. Jaynes, Information Theory and Statistical Mechanics. II, Phys. Rev. 108 (1957) 171 [INSPIRE].
R. Bousso, V. Chandrasekaran and A. Shahbazi-Moghaddam, From black hole entropy to energy-minimizing states in QFT, Phys. Rev. D 101 (2020) 046001 [arXiv:1906.05299] [INSPIRE].
N. Engelhardt, G. Penington and A. Shahbazi-Moghaddam, Twice upon a time: timelike-separated quantum extremal surfaces, JHEP 01 (2024) 033 [arXiv:2308.16226] [INSPIRE].
S. Arora and B. Barak, Computational complexity: a modern approach, Cambridge University Press (2009).
A. Almheiri and H.W. Lin, The entanglement wedge of unknown couplings, JHEP 08 (2022) 062 [arXiv:2111.06298] [INSPIRE].
D. Harlow and J.-Q. Wu, Algebra of diffeomorphism-invariant observables in Jackiw-Teitelboim gravity, JHEP 05 (2022) 097 [arXiv:2108.04841] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
N. Engelhardt and Å. Folkestad, Canonical purification of evaporating black holes, Phys. Rev. D 105 (2022) 086010 [arXiv:2201.08395] [INSPIRE].
Y. Zhao, Collision in the interior of wormhole, JHEP 03 (2020) 144 [arXiv:2011.06016] [INSPIRE].
F.M. Haehl and Y. Zhao, Diagnosing collisions in the interior of a wormhole, Phys. Rev. D 104 (2021) L021901 [arXiv:2104.02736] [INSPIRE].
F.M. Haehl and Y. Zhao, Collisions of localized shocks and quantum circuits, JHEP 09 (2022) 002 [arXiv:2202.04661] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [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].
D. Harlow and E. Shaghoulian, Global symmetry, Euclidean gravity, and the black hole information problem, JHEP 04 (2021) 175 [arXiv:2010.10539] [INSPIRE].
S.R. Coleman, Black holes as red herrings: Topological fluctuations and the loss of quantum coherence, Nucl. Phys. B 307 (1988) 867 [INSPIRE].
S.B. Giddings and A. Strominger, Loss of incoherence and determination of coupling constants in quantum gravity, Nucl. Phys. B 307 (1988) 854 [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
Y. Aharonov, P.G. Bergmann and J.L. Lebowitz, Time Symmetry in the Quantum Process of Measurement, Phys. Rev. 134 (1964) B1410 [INSPIRE].
D. Gottesman and J. Preskill, Comment on ‘The Black hole final state’, JHEP 03 (2004) 026 [hep-th/0311269] [INSPIRE].
S. Lloyd and J. Preskill, Unitarity of black hole evaporation in final-state projection models, JHEP 08 (2014) 126 [arXiv:1308.4209] [INSPIRE].
R. Bousso and D. Stanford, Measurements without Probabilities in the Final State Proposal, Phys. Rev. D 89 (2014) 044038 [arXiv:1310.7457] [INSPIRE].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
K. Papadodimas and S. Raju, Black Hole Interior in the Holographic Correspondence and the Information Paradox, Phys. Rev. Lett. 112 (2014) 051301 [arXiv:1310.6334] [INSPIRE].
D. Harlow, Aspects of the Papadodimas-Raju Proposal for the Black Hole Interior, JHEP 11 (2014) 055 [arXiv:1405.1995] [INSPIRE].
D. Marolf and J. Polchinski, Violations of the Born rule in cool state-dependent horizons, JHEP 01 (2016) 008 [arXiv:1506.01337] [INSPIRE].
I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS2 gravity, arXiv:1707.02325 [INSPIRE].
C. Akers, N. Engelhardt and D. Harlow, Simple holographic models of black hole evaporation, JHEP 08 (2020) 032 [arXiv:1910.00972] [INSPIRE].
V. Balasubramanian, A. Kar, C. Li and O. Parrikar, Quantum error correction in the black hole interior, JHEP 07 (2023) 189 [arXiv:2203.01961] [INSPIRE].
T. Anous, J. Kruthoff and R. Mahajan, Density matrices in quantum gravity, SciPost Phys. 9 (2020) 045 [arXiv:2006.17000] [INSPIRE].
K. Langhoff and Y. Nomura, Ensemble from Coarse Graining: Reconstructing the Interior of an Evaporating Black Hole, Phys. Rev. D 102 (2020) 086021 [arXiv:2008.04202] [INSPIRE].
A. Blommaert, L.V. Iliesiu and J. Kruthoff, Alpha states demystified — towards microscopic models of AdS2 holography, JHEP 08 (2022) 071 [arXiv:2203.07384] [INSPIRE].
R. Chao, B.W. Reichardt, C. Sutherland and T. Vidick, Overlapping qubits, arXiv:1701.01062 [https://doi.org/10.4230/LIPIcs.ITCS.2017.48].
H. Buhrman, R. Cleve, J. Watrous and R. de Wolf, Quantum Fingerprinting, Phys. Rev. Lett. 87 (2001) 167902 [quant-ph/0102001] [INSPIRE].
R. Bousso, Violations of the Equivalence Principle by a Nonlocally Reconstructed Vacuum at the Black Hole Horizon, Phys. Rev. Lett. 112 (2014) 041102 [arXiv:1308.3697] [INSPIRE].
M. Van Raamsdonk, Evaporating Firewalls, JHEP 11 (2014) 038 [arXiv:1307.1796] [INSPIRE].
B. Guo, M.R.R. Hughes, S.D. Mathur and M. Mehta, Contrasting the fuzzball and wormhole paradigms for black holes, Turk. J. Phys. 45 (2021) 281 [arXiv:2111.05295] [INSPIRE].
R. Bousso and A. Shahbazi-Moghaddam, Quantum singularities, Phys. Rev. D 107 (2023) 066002 [arXiv:2206.07001] [INSPIRE].
D. Harlow and L. Susskind, Crunches, Hats, and a Conjecture, arXiv:1012.5302 [INSPIRE].
B. Collins, Moments and cumulants of polynomial random variables on unitarygroups, the itzykson-zuber integral, and free probability, Int. Math. Res. Not. 2003 (2003) 953.
B. Collins and P. Śniady, Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group, Commun. Math. Phys. 264 (2006) 773 [INSPIRE].
W. Fulton and J. Harris, Representation theory: a first course, vol. 129, Springer Science & Business Media (2013).
C. Dankert, R. Cleve, J. Emerson and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80 (2009) 012304 [INSPIRE].
G.W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices, Cambridge university press (2010).
M. Émery and J.E. Yukich, A simple proof of the logarithmic sobolev inequality on the circle, Séminaire de probabilités de Strasbourg 21 (1987) 173.
M. Ledoux, The concentration of measure phenomenon, American Mathematical Soc. (2001).
D. Bakry and M. Émery, Diffusions hypercontractives, in Seminaire de probabilités XIX 1983/84, Springer (1985) pp. 177–206.
M. Gromov and V.D. Milman, A topological application of the isoperimetric inequality, Am. J. MAth. 105 (1983) 843.
S.J. Szarek, Spaces with large distance to \( {\ell}_{\infty}^n \) and random matrices, Am. J. MAth. 112 (1990) 899.
M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th ed., Cambridge University Press, U.S.A. (2011).
A. Nahum, S. Vijay and J. Haah, Operator Spreading in Random Unitary Circuits, Phys. Rev. X 8 (2018) 021014 [arXiv:1705.08975] [INSPIRE].
C. von Keyserlingk, T. Rakovszky, F. Pollmann and S. Sondhi, Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8 (2018) 021013 [arXiv:1705.08910] [INSPIRE].
Acknowledgments
We thank Scott Aaronson, Ahmed Almheiri, Raphael Bousso, Wissam Chemissany, Patrick Hayden, Hong Liu, Juan Maldacena, Don Marolf, Henry Maxfield, Yasunori Nomura, and Herman Verlinde for useful discussions. CA is supported by the Simons Foundation as an “It from Qubit” fellow, the Air Force Office of Scientific Research under the award number FA9550-19-1-0360, the US Department of Energy under grant DE-SC0012567, the John Templeton Foundation and the Gordon and Betty Moore Foundation via the Black Hole Initiative, and the National Science Foundation under grant no. PHY-2011905. NE is supported in part by NSF grant no. PHY-2011905, by the U.S. Department of Energy Early Career Award DE-SC0021886, by the John Templeton Foundation and the Gordon and Betty Moore Foundation via the Black Hole Initiative, and by funds from the MIT department of physics. DH is supported by the Simons Foundation as a member of the “It from Qubit” collaboration, the Sloan Foundation as a Sloan Fellow, the Packard Foundation as a Packard Fellow, the Air Force Office of Scientific Research under the award number FA9550-19-1-0360, the US Department of Energy under grants DE-SC0012567 and DE-SC0020360, and the MIT department of physics. GP is supported by the UC Berkeley Physics Department, the Simons Foundation through the “It from Qubit” program, the Department of Energy via the GeoFlow consortium (QuantISED Award DE-SC0019380), and AFOSR award FA9550-22-1-0098. He also acknowledges support from an IBM Einstein Fellowship at the Institute for Advanced Study. SV is supported by US Department of Energy under grants DE-SC0012567 and DE-SC002036.
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Akers, C., Engelhardt, N., Harlow, D. et al. The black hole interior from non-isometric codes and complexity. J. High Energ. Phys. 2024, 155 (2024). https://doi.org/10.1007/JHEP06(2024)155
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DOI: https://doi.org/10.1007/JHEP06(2024)155