Abstract
We study Renyi entropies Sn in quantum error correcting codes and compare the answer to the cosmic brane prescription for computing \( {\tilde{S}}_n\equiv {n}^2{\partial}_n\left(\frac{n-1}{n}{S}_n\right) \). We find that general operator algebra codes have a similar, more general prescription. Notably, for the AdS/CFT code to match the specific cosmic brane prescription, the code must have maximal entanglement within eigenspaces of the area operator. This gives us an improved definition of the area operator, and establishes a stronger connection between the Ryu-Takayanagi area term and the edge modes in lattice gauge theory. We also propose a new interpretation of existing holographic tensor networks as area eigenstates instead of smooth geometries. This interpretation would explain why tensor networks have historically had trouble modeling the Renyi entropy spectrum of holographic CFTs, and it suggests a method to construct holographic networks with the correct spectrum.
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References
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].
J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle and M. Walter, Entanglement Wedge Reconstruction via Universal Recovery Channels, arXiv:1704.05839 [INSPIRE].
P. Hayden and G. Penington, Learning the Alpha-bits of Black Holes, arXiv:1807.06041 [INSPIRE].
A. Almheiri, Holographic Quantum Error Correction and the Projected Black Hole Interior, arXiv:1810.02055 [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
X. Dong, The Gravity Dual of Renyi Entropy, Nature Commun. 7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].
A. Almheiri, X. Dong and B. Swingle, Linearity of Holographic Entanglement Entropy, JHEP 02 (2017) 074 [arXiv:1606.04537] [INSPIRE].
X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations and the Equations of Motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [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, 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].
X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, arXiv:1811.05382 [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].
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].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
D.L. Jafferis and S.J. Suh, The Gravity Duals of Modular Hamiltonians, JHEP 09 (2016) 068 [arXiv:1412.8465] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP 07 (2017) 151 [arXiv:1704.05464] [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, Holographic Entanglement Entropy for General Higher Derivative Gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].
W. Donnelly, B. Michel, D. Marolf and J. Wien, Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers, JHEP 04 (2017) 093 [arXiv:1611.05841] [INSPIRE].
W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].
W. Donnelly and A.C. Wall, Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D 94 (2016) 104053 [arXiv:1506.05792] [INSPIRE].
J. Lin, Ryu-Takayanagi Area as an Entanglement Edge Term, arXiv:1704.07763 [INSPIRE].
D.L. Jafferis, Bulk reconstruction and the Hartle-Hawking wavefunction, arXiv:1703.01519 [INSPIRE].
J.B. Hartle and S.W. Hawking, Wave Function of the Universe, Phys. Rev. D 28 (1983) 2960 [INSPIRE].
X.-L. Qi, Z. Yang and Y.-Z. You, Holographic coherent states from random tensor networks, JHEP 08 (2017) 060 [arXiv:1703.06533] [INSPIRE].
M. Han and S. Huang, Discrete gravity on random tensor network and holographic Rényi entropy, JHEP 11 (2017) 148 [arXiv:1705.01964] [INSPIRE].
D. Harlow, Wormholes, Emergent Gauge Fields and the Weak Gravity Conjecture, JHEP 01 (2016) 122 [arXiv:1510.07911] [INSPIRE].
A.J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, JHEP 02 (2018) 021 [arXiv:1706.05061] [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, arXiv:1806.01281 [INSPIRE].
Y. Nomura and G.N. Remmen, Area Law Unification and the Holographic Event Horizon, JHEP 08 (2018) 063 [arXiv:1805.09339] [INSPIRE].
Y. Nomura, N. Salzetta, F. Sanches and S.J. Weinberg, Toward a Holographic Theory for General Spacetimes, Phys. Rev. D 95 (2017) 086002 [arXiv:1611.02702] [INSPIRE].
Y. Nomura, P. Rath and N. Salzetta, Classical Spacetimes as Amplified Information in Holographic Quantum Theories, Phys. Rev. D 97 (2018) 106025 [arXiv:1705.06283] [INSPIRE].
Y. Nomura, P. Rath and N. Salzetta, Spacetime from Unentanglement, Phys. Rev. D 97 (2018) 106010 [arXiv:1711.05263] [INSPIRE].
Y. Nomura, P. Rath and N. Salzetta, Pulling the Boundary into the Bulk, Phys. Rev. D 98 (2018) 026010 [arXiv:1805.00523] [INSPIRE].
M. Alishahiha, A. Karch, E. Silverstein and D. Tong, The dS/dS correspondence, AIP Conf. Proc. 743 (2004) 393 [hep-th/0407125] [INSPIRE].
X. Dong, E. Silverstein and G. Torroba, de Sitter Holography and Entanglement Entropy, JHEP 07 (2018) 050 [arXiv:1804.08623] [INSPIRE].
M. Miyaji, T. Takayanagi and K. Watanabe, From path integrals to tensor networks for the AdS/CFT correspondence, Phys. Rev. D 95 (2017) 066004 [arXiv:1609.04645] [INSPIRE].
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Continuous Multiscale Entanglement Renormalization Ansatz as Holographic Surface-State Correspondence, Phys. Rev. Lett. 115 (2015) 171602 [arXiv:1506.01353] [INSPIRE].
F. Sanches and S.J. Weinberg, Holographic entanglement entropy conjecture for general spacetimes, Phys. Rev. D 94 (2016) 084034 [arXiv:1603.05250] [INSPIRE].
M. Miyaji and T. Takayanagi, Surface/State Correspondence as a Generalized Holography, PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].
W. Donnelly and V. Shyam, Entanglement entropy and \( T\overline{T} \) deformation, Phys. Rev. Lett. 121 (2018) 131602 [arXiv:1806.07444] [INSPIRE].
M. Headrick and T. Takayanagi, A Holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].
N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully and M. Walter, The Holographic Entropy Cone, JHEP 09 (2015) 130 [arXiv:1505.07839] [INSPIRE].
N. Bao, C. Cao, M. Walter and Z. Wang, Holographic entropy inequalities and gapped phases of matter, JHEP 09 (2015) 203 [arXiv:1507.05650] [INSPIRE].
G. Camilo, G.T. Landi and S. Eliëns, Strong subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states, Phys. Rev. B 99 (2019) 045155 [arXiv:1810.07070] [INSPIRE].
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Akers, C., Rath, P. Holographic Renyi entropy from quantum error correction. J. High Energ. Phys. 2019, 52 (2019). https://doi.org/10.1007/JHEP05(2019)052
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DOI: https://doi.org/10.1007/JHEP05(2019)052