Abstract
We analyze the error correcting properties of the Sachdev-Ye-Kitaev model, with errors that correspond to erasures of subsets of fermions. We study the limit where the number of fermions erased is large but small compared to the total number of fermions. We compute the price of the quantum error correcting code, defined as the number of physical qubits needed to reconstruct whether a given operator has been acted upon the thermal state or not. By thinking about reconstruction via quantum teleportation, we argue for a bound that relates the price to the ordinary operator size in systems that display so-called detailed size winding [1]. We then find that in SYK the price roughly saturates this bound. Computing the price requires computing modular flowed correlators with respect to the density matrix associated to a subset of fermions. We offer an interpretation of these correlators as probing a quantum extremal surface in the AdS dual of SYK. In the large N limit, the operator algebras associated to subsets of fermions in SYK satisfy half-sided modular inclusion, which is indicative of an emergent Type III1 von Neumann algebra. We discuss the relationship between the emergent algebra of half-sided modular inclusions and bulk symmetry generators.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Nezami, H.W. Lin, A.R. Brown, H. Gharibyan, S. Leichenauer, G. Salton et al., Quantum Gravity in the Lab: Teleportation by Size and Traversable Wormholes, Part II, arXiv:2102.01064 [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].
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].
L. Susskind, Complexity and Newton’s Laws, Front. in Phys. 8 (2020) 262 [arXiv:1904.12819] [INSPIRE].
A.R. Brown, H. Gharibyan, A. Streicher, L. Susskind, L. Thorlacius and Y. Zhao, Falling Toward Charged Black Holes, Phys. Rev. D 98 (2018) 126016 [arXiv:1804.04156] [INSPIRE].
F.M. Haehl and Y. Zhao, Size and momentum of an infalling particle in the black hole interior, JHEP 06 (2021) 056 [arXiv:2102.05697] [INSPIRE].
X.-L. Qi and A. Streicher, Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK, JHEP 08 (2019) 012 [arXiv:1810.11958] [INSPIRE].
T. Schuster, B. Kobrin, P. Gao, I. Cong, E.T. Khabiboulline, N.M. Linke et al., Many-body quantum teleportation via operator spreading in the traversable wormhole protocol, arXiv:2102.00010 [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
F. Pastawski and J. Preskill, Code properties from holographic geometries, Phys. Rev. X 7 (2017) 021022 [arXiv:1612.00017] [INSPIRE].
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].
A.R. Brown, H. Gharibyan, S. Leichenauer, H.W. Lin, S. Nezami, G. Salton et al., Quantum Gravity in the Lab: Teleportation by Size and Traversable Wormholes, arXiv:1911.06314 [INSPIRE].
V. Chandrasekaran, T. Faulkner and A. Levine, Scattering strings off quantum extremal surfaces, arXiv:2108.01093 [INSPIRE].
J. De Boer and L. Lamprou, Holographic Order from Modular Chaos, JHEP 06 (2020) 024 [arXiv:1912.02810] [INSPIRE].
H.W. Lin, J. Maldacena and Y. Zhao, Symmetries Near the Horizon, JHEP 08 (2019) 049 [arXiv:1904.12820] [INSPIRE].
S. Leutheusser and H. Liu, Emergent times in holographic duality, arXiv:2112.12156 [INSPIRE].
E. Witten, Gravity and the Crossed Product, arXiv:2112.12828 [INSPIRE].
P. Gao, D.L. Jafferis and A.C. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
A. Levine, A. Shahbazi-Moghaddam and R.M. Soni, Seeing the entanglement wedge, JHEP 06 (2021) 134 [arXiv:2009.11305] [INSPIRE].
A. May and E. Hijano, The holographic entropy zoo, JHEP 10 (2018) 036 [arXiv:1806.06077] [INSPIRE].
Y. Suzuki, T. Takayanagi and K. Umemoto, Entanglement Wedges from the Information Metric in Conformal Field Theories, Phys. Rev. Lett. 123 (2019) 221601 [arXiv:1908.09939] [INSPIRE].
Y. Kusuki, Y. Suzuki, T. Takayanagi and K. Umemoto, Looking at Shadows of Entanglement Wedges, PTEP 2020 (2020) 11B105 [arXiv:1912.08423] [INSPIRE].
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A General Proof of the Quantum Null Energy Condition, JHEP 09 (2019) 020 [arXiv:1706.09432] [INSPIRE].
T. Faulkner, M. Li and H. Wang, A modular toolkit for bulk reconstruction, JHEP 04 (2019) 119 [arXiv:1806.10560] [INSPIRE].
P. Zhang, C. Liu and X. Chen, Subsystem Rényi Entropy of Thermal Ensembles for SYK-like models, SciPost Phys. 8 (2020) 094 [arXiv:2003.09766] [INSPIRE].
T. Faulkner, Bulk Emergence and the RG Flow of Entanglement Entropy, JHEP 05 (2015) 033 [arXiv:1412.5648] [INSPIRE].
S. Balakrishnan, V. Chandrasekaran, T. Faulkner, A. Levine and A. Shahbazi-Moghaddam, Entropy Variations and Light Ray Operators from Replica Defects, arXiv:1906.08274 [INSPIRE].
S. Balakrishnan, S. Dutta and T. Faulkner, Gravitational dual of the Rényi twist displacement operator, Phys. Rev. D 96 (2017) 046019 [arXiv:1607.06155] [INSPIRE].
A. Streicher, SYK Correlators for All Energies, JHEP 02 (2020) 048 [arXiv:1911.10171] [INSPIRE].
A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [INSPIRE].
C. Akers, J. Koeller, S. Leichenauer and A. Levine, Geometric Constraints from Subregion Duality Beyond the Classical Regime, arXiv:1610.08968 [INSPIRE].
J. Koeller and S. Leichenauer, Holographic Proof of the Quantum Null Energy Condition, Phys. Rev. D 94 (2016) 024026 [arXiv:1512.06109] [INSPIRE].
C. Agón and T. Faulkner, Quantum Corrections to Holographic Mutual Information, JHEP 08 (2016) 118 [arXiv:1511.07462] [INSPIRE].
X.-L. Qi, Z. Shangnan and Z. Yang, Holevo information and ensemble theory of gravity, JHEP 02 (2022) 056 [arXiv:2111.05355] [INSPIRE].
S. Leutheusser and H. Liu, Causal connectability between quantum systems and the black hole interior in holographic duality, arXiv:2110.05497 [INSPIRE].
Y. Gu, A. Kitaev and P. Zhang, A two-way approach to out-of-time-order correlators, JHEP 03 (2022) 133 [arXiv:2111.12007] [INSPIRE].
J. Maldacena and A. Milekhin, To gauge or not to gauge?, JHEP 04 (2018) 084 [arXiv:1802.00428] [INSPIRE].
A. Milekhin, Quantum error correction and large N, SciPost Phys. 11 (2021) 094 [arXiv:2008.12869] [INSPIRE].
T. Anous, J.L. Karczmarek, E. Mintun, M. Van Raamsdonk and B. Way, Areas and entropies in BFSS/gravity duality, SciPost Phys. 8 (2020) 057 [arXiv:1911.11145] [INSPIRE].
M. Beccaria, H. Jiang and A.A. Tseytlin, Supersymmetric Liouville theory in AdS2 and AdS/CFT, JHEP 11 (2019) 051 [arXiv:1909.10255] [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: 2203.05058
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
Chandrasekaran, V., Levine, A. Quantum error correction in SYK and bulk emergence. J. High Energ. Phys. 2022, 39 (2022). https://doi.org/10.1007/JHEP06(2022)039
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2022)039