Abstract
It is conventional to study the entanglement between spatial regions of a quantum field theory. However, in some systems entanglement can be dominated by “internal”, possibly gauged, degrees of freedom that are not spatially organized, and that can give rise to gaps smaller than the inverse size of the system. In a holographic context, such small gaps are associated to the appearance of horizons and singularities in the dual spacetime. Here, we propose a concept of entwinement, which is intended to capture this fine structure of the wavefunction. Holographically, entwinement probes the entanglement shadow — the region of spacetime not probed by the minimal surfaces that compute spatial entanglement in the dual field theory. We consider the simplest example of this scenario — a 2d conformal field theory (CFT) that is dual to a conical defect in AdS3 space. Following our previous work, we show that spatial entanglement in the CFT reproduces spacetime geometry up to a finite distance from the conical defect. We then show that the interior geometry up to the defect can be reconstructed from entwinement that is sensitive to the discretely gauged, fractionated degrees of freedom of the CFT. Entwinement in the CFT is related to non-minimal geodesics in the conical defect geometry, suggesting a potential quantum information theoretic meaning for these objects in a holographic context. These results may be relevant for the reconstruction of black hole interiors from a dual field theory.
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References
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].
V. Balasubramanian, B. Czech, B.D. Chowdhury and J. de Boer, The entropy of a hole in spacetime, JHEP 10 (2013) 220 [arXiv:1305.0856] [INSPIRE].
V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, Bulk curves from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204] [INSPIRE].
R.C. Myers, J. Rao and S. Sugishita, Holographic holes in higher dimensions, JHEP 06 (2014) 044 [arXiv:1403.3416] [INSPIRE].
B. Czech, X. Dong and J. Sully, Holographic reconstruction of general bulk surfaces, JHEP 11 (2014) 015 [arXiv:1406.4889] [INSPIRE].
M. Headrick, R.C. Myers and J. Wien, Holographic holes and differential entropy, JHEP 10 (2014) 149 [arXiv:1408.4770] [INSPIRE].
E. Bianchi and R.C. Myers, On the architecture of spacetime geometry, Class. Quant. Grav. 31 (2014) 214002 [arXiv:1212.5183] [INSPIRE].
J. Hammersley, Extracting the bulk metric from boundary information in asymptotically AdS spacetimes, JHEP 12 (2006) 047 [hep-th/0609202] [INSPIRE].
S. Bilson, Extracting spacetimes using the AdS/CFT conjecture, JHEP 08 (2008) 073 [arXiv:0807.3695] [INSPIRE].
S. Bilson, Extracting spacetimes using the AdS/CFT conjecture. Part II, JHEP 02 (2011) 050 [arXiv:1012.1812] [INSPIRE].
M. Spillane, Constructing space from entanglement entropy, arXiv:1311.4516 [INSPIRE].
B. Czech and L. Lamprou, Holographic definition of points and distances, Phys. Rev. D 90 (2014) 106005 [arXiv:1409.4473] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [Int. J. Mod. Phys. D 19 (2010) 2429] [arXiv:1005.3035] [INSPIRE].
B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortschr. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, arXiv:1408.6300 [INSPIRE].
V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
A.W. Peet and J. Polchinski, UV/IR relations in AdS dynamics, Phys. Rev. D 59 (1999) 065011 [hep-th/9809022] [INSPIRE].
V. Balasubramanian, S.B. Giddings and A.E. Lawrence, What do CFTs tell us about anti-de Sitter space-times?, JHEP 03 (1999) 001 [hep-th/9902052] [INSPIRE].
S.B. Giddings, Black hole information, unitarity and nonlocality, Phys. Rev. D 74 (2006) 106005 [hep-th/0605196] [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, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, AdS field theory from conformal field theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [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].
L. Susskind and E. Witten, The holographic bound in anti-de Sitter space, hep-th/9805114 [INSPIRE].
J.M. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475 (1996) 679 [hep-th/9604042] [INSPIRE].
R. Bousso and A.L. Mints, Holography and entropy bounds in the plane wave matrix model, Phys. Rev. D 73 (2006) 126005 [hep-th/0512201] [INSPIRE].
H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [INSPIRE].
D. Berenstein, Large-N BPS states and emergent quantum gravity, JHEP 01 (2006) 125 [hep-th/0507203] [INSPIRE].
V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, The library of Babel: on the origin of gravitational thermodynamics, JHEP 12 (2005) 006 [hep-th/0508023] [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
L. Susskind, Some speculations about black hole entropy in string theory, in C. Teitelboim ed., The black hole, pp. 118–131 [hep-th/9309145] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
S.R. Das and S.D. Mathur, Excitations of D strings, entropy and duality, Phys. Lett. B 375 (1996) 103 [hep-th/9601152] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.W. Peet, Entropy and temperature of black 3-branes, Phys. Rev. D 54 (1996) 3915 [hep-th/9602135] [INSPIRE].
V. Balasubramanian, M.B. McDermott and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory, Phys. Rev. D 86 (2012) 045014 [arXiv:1108.3568] [INSPIRE].
E. Verlinde and H. Verlinde, Black hole entanglement and quantum error correction, JHEP 10 (2013) 107 [arXiv:1211.6913] [INSPIRE].
E. Verlinde and H. Verlinde, Behind the horizon in AdS/CFT, arXiv:1311.1137 [INSPIRE].
E. Keski-Vakkuri, Bulk and boundary dynamics in BTZ black holes, Phys. Rev. D 59 (1999) 104001 [hep-th/9808037] [INSPIRE].
V. Balasubramanian, P. Kraus and M. Shigemori, Massless black holes and black rings as effective geometries of the D1-D5 system, Class. Quant. Grav. 22 (2005) 4803 [hep-th/0508110] [INSPIRE].
V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev. D 61 (2000) 044007 [hep-th/9906226] [INSPIRE].
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
J. de Boer, M.M. Sheikh-Jabbari and J. Simon, Near horizon limits of massless BTZ and their CFT duals, Class. Quant. Grav. 28 (2011) 175012 [arXiv:1011.1897] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for M N /S N orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
E.J. Martinec and W. McElgin, String theory on AdS orbifolds, JHEP 04 (2002) 029 [hep-th/0106171] [INSPIRE].
E.J. Martinec and W. McElgin, Exciting AdS orbifolds, JHEP 10 (2002) 050 [hep-th/0206175] [INSPIRE].
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
C.A. Agon, M. Headrick, D.L. Jafferis and S. Kasko, Disk entanglement entropy for a Maxwell field, Phys. Rev. D 89 (2014) 025018 [arXiv:1310.4886] [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].
K. Ohmori and Y. Tachikawa, Physics at the entangling surface, arXiv:1406.4167 [INSPIRE].
V.E. Hubeny, Covariant residual entropy, JHEP 09 (2014) 156 [arXiv:1406.4611] [INSPIRE].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2+1) black hole, Phys. Rev. D 48 (1993) 1506 [gr-qc/9302012] [INSPIRE].
H. Araki and E.H. Lieb, Entropy inequalities, Commun. Math. Phys. 18 (1970) 160 [INSPIRE].
B.D. Chowdhury, Limitations of holography, arXiv:1405.4292 [INSPIRE].
B.S. Kay, Instability of enclosed horizons, arXiv:1310.7395 [INSPIRE].
S.G. Avery and B.D. Chowdhury, No holography for eternal AdS black holes, arXiv:1312.3346 [INSPIRE].
S.D. Mathur, What is the dual of two entangled CFTs?, arXiv:1402.6378 [INSPIRE].
D. Kabat and G. Lifschytz, Finite N and the failure of bulk locality: black holes in AdS/CFT, JHEP 09 (2014) 077 [arXiv:1405.6394] [INSPIRE].
J. Erdmenger, M. Flory and C. Sleight, Conditions on holographic entangling surfaces in higher curvature gravity, JHEP 06 (2014) 104 [arXiv:1401.5075] [INSPIRE].
F. Nogueira, Extremal surfaces in asymptotically AdS charged boson stars backgrounds, Phys. Rev. D 87 (2013) 106006 [arXiv:1301.4316] [INSPIRE].
S.A. Gentle and M. Rangamani, Holographic entanglement and causal information in coherent states, JHEP 01 (2014) 120 [arXiv:1311.0015] [INSPIRE].
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Balasubramanian, V., Chowdhury, B.D., Czech, B. et al. Entwinement and the emergence of spacetime. J. High Energ. Phys. 2015, 48 (2015). https://doi.org/10.1007/JHEP01(2015)048
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DOI: https://doi.org/10.1007/JHEP01(2015)048