We define a generalized entanglement measure in the context of the AdS/CFT correspondence. Compared to the ordinary entanglement entropy for a spatial subregion dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary theory. Moreover, we resolve the contribution to the entanglement entropy of strings with different winding numbers in the bulk geometry. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including the entanglement shadow up to the horizon in the one-sided black hole as well as the wormhole growth in the case of a two-sided black hole for an arbitrarily long time. Therefore, we propose that the entanglement structure of the boundary state is enough to reconstruct asymptotically AdS3 geometries up to extremal surface barriers.
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, Class. Quant. Grav. 31 (2014) 214002 [arXiv:1212.5183] [INSPIRE].
V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].
V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].
B. Freivogel, R. Jefferson, L. Kabir, B. Mosk and I.-S. Yang, Casting Shadows on Holographic Reconstruction, Phys. Rev. D 91 (2015) 086013 [arXiv:1412.5175] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [Addendum ibid. 64 (2016) 44] [arXiv:1403.5695] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
V. Balasubramanian, A. Bernamonti, B. Craps, T. De Jonckheere and F. Galli, Entwinement in discretely gauged theories, JHEP 12 (2016) 094 [arXiv:1609.03991] [INSPIRE].
V. Balasubramanian, B. Craps, T. De Jonckheere and G. Sárosi, Entanglement versus entwinement in symmetric product orbifolds, JHEP 01 (2019) 190 [arXiv:1806.02871] [INSPIRE].
J. Erdmenger and M. Gerbershagen, Entwinement as a possible alternative to complexity, JHEP 03 (2020) 082 [arXiv:1910.05352] [INSPIRE].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
M. Gerbershagen, Monodromy methods for torus conformal blocks and entanglement entropy at large central charge, JHEP 08 (2021) 143 [arXiv:2101.11642] [INSPIRE].
T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [INSPIRE].
G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS3 at ‖ = 1, JHEP 08 (2018) 204 [arXiv:1803.04420] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS3, JHEP 05 (2018) 085 [arXiv:1803.04423] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the Symmetric Product CFT, JHEP 04 (2019) 103 [arXiv:1812.01007] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, Deriving the AdS3/CFT2 correspondence, JHEP 02 (2020) 136 [arXiv:1911.00378] [INSPIRE].
A. Dei, M.R. Gaberdiel, R. Gopakumar and B. Knighton, Free field world-sheet correlators for AdS3, JHEP 02 (2021) 081 [arXiv:2009.11306] [INSPIRE].
L. Eberhardt, Partition functions of the tensionless string, JHEP 03 (2021) 176 [arXiv:2008.07533] [INSPIRE].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Deforming the D1D5 CFT away from the orbifold point, JHEP 06 (2010) 031 [arXiv:1002.3132] [INSPIRE].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Excitations in the deformed D1D5 CFT, JHEP 06 (2010) 032 [arXiv:1003.2746] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, A Spin Chain for the Symmetric Product CFT2, JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].
C.T. Asplund and S.G. Avery, Evolution of Entanglement Entropy in the D1-D5 Brane System, Phys. Rev. D 84 (2011) 124053 [arXiv:1108.2510] [INSPIRE].
B.A. Burrington, A.W. Peet and I.G. Zadeh, Operator mixing for string states in the D1-D5 CFT near the orbifold point, Phys. Rev. D 87 (2013) 106001 [arXiv:1211.6699] [INSPIRE].
Z. Carson, S. Hampton, S.D. Mathur and D. Turton, Effect of the deformation operator in the D1D5 CFT, JHEP 01 (2015) 071 [arXiv:1410.4543] [INSPIRE].
Z. Carson, S. Hampton, S.D. Mathur and D. Turton, Effect of the twist operator in the D1D5 CFT, JHEP 08 (2014) 064 [arXiv:1405.0259] [INSPIRE].
Z. Carson, S.D. Mathur and D. Turton, Bogoliubov coefficients for the twist operator in the D1D5 CFT, Nucl. Phys. B 889 (2014) 443 [arXiv:1406.6977] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, Second order effect of twist deformations in the D1D5 CFT, JHEP 04 (2016) 115 [arXiv:1511.04046] [INSPIRE].
M.R. Gaberdiel, C. Peng and I.G. Zadeh, Higgsing the stringy higher spin symmetry, JHEP 10 (2015) 101 [arXiv:1506.02045] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, One-Loop Transition Amplitudes in the D1D5 CFT, JHEP 01 (2017) 006 [arXiv:1606.06212] [INSPIRE].
Z. Carson, S. Hampton and S.D. Mathur, Full action of two deformation operators in the D1D5 CFT, JHEP 11 (2017) 096 [arXiv:1612.03886] [INSPIRE].
B.A. Burrington, I.T. Jardine and A.W. Peet, Operator mixing in deformed D1D5 CFT and the OPE on the cover, JHEP 06 (2017) 149 [arXiv:1703.04744] [INSPIRE].
S. Hampton, S.D. Mathur and I.G. Zadeh, Lifting of D1-D5-P states, JHEP 01 (2019) 075 [arXiv:1804.10097] [INSPIRE].
B. Guo and S.D. Mathur, Lifting of states in 2-dimensional N = 4 supersymmetric CFTs, JHEP 10 (2019) 155 [arXiv:1905.11923] [INSPIRE].
B. Guo and S.D. Mathur, Lifting of level-1 states in the D1D5 CFT, JHEP 03 (2020) 028 [arXiv:1912.05567] [INSPIRE].
B. Guo and S.D. Mathur, Lifting at higher levels in the D1D5 CFT, JHEP 11 (2020) 145 [arXiv:2008.01274] [INSPIRE].
A.A. Lima, G.M. Sotkov and M. Stanishkov, On the Dynamics of Protected Ramond Ground States in the D1-D5 CFT, arXiv:2103.04459 [INSPIRE].
B. Czech, X. Dong and J. Sully, Holographic Reconstruction of General Bulk Surfaces, JHEP 11 (2014) 015 [arXiv:1406.4889] [INSPIRE].
B. Czech and L. Lamprou, Holographic definition of points and distances, Phys. Rev. D 90 (2014) 106005 [arXiv:1409.4473] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
C.T. Asplund, N. Callebaut and C. Zukowski, Equivalence of Emergent de Sitter Spaces from Conformal Field Theory, JHEP 09 (2016) 154 [arXiv:1604.02687] [INSPIRE].
J.-d. Zhang and B. Chen, Kinematic Space and Wormholes, JHEP 01 (2017) 092 [arXiv:1610.07134] [INSPIRE].
J.C. Cresswell and A.W. Peet, Kinematic space for conical defects, JHEP 11 (2017) 155 [arXiv:1708.09838] [INSPIRE].
R. Abt, J. Erdmenger, M. Gerbershagen, C.M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, JHEP 01 (2019) 012 [arXiv:1805.10298] [INSPIRE].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
M. Headrick and V.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav. 35 (2018) 10 [arXiv:1710.09516] [INSPIRE].
S.X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, Bit Threads and Holographic Monogamy, Commun. Math. Phys. 376 (2019) 609 [arXiv:1808.05234] [INSPIRE].
C.A. Agón, E. Cáceres and J.F. Pedraza, Bit threads, Einstein’s equations and bulk locality, JHEP 01 (2021) 193 [arXiv:2007.07907] [INSPIRE].
N. Engelhardt and A.C. Wall, Extremal Surface Barriers, JHEP 03 (2014) 068 [arXiv:1312.3699] [INSPIRE].
M. Goldstein and E. Sela, Symmetry-resolved entanglement in many-body systems, Phys. Rev. Lett. 120 (2018) 200602 [arXiv:1711.09418] [INSPIRE].
J.C. Xavier, F.C. Alcaraz and G. Sierra, Equipartition of the entanglement entropy, Phys. Rev. B 98 (2018) 041106 [arXiv:1804.06357] [INSPIRE].
R. Bonsignori, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement in free fermionic systems, J. Phys. A 52 (2019) 475302 [arXiv:1907.02084] [INSPIRE].
H. Barghathi, E. Casiano-Diaz and A. Del Maestro, Operationally accessible entanglement of one-dimensional spinless fermions, Phys. Rev. A 100 (2019) 022324 [arXiv:1905.03312] [INSPIRE].
N. Feldman and M. Goldstein, Dynamics of Charge-Resolved Entanglement after a Local Quench, Phys. Rev. B 100 (2019) 235146 [arXiv:1905.10749] [INSPIRE].
S. Fraenkel and M. Goldstein, Symmetry resolved entanglement: Exact results in 1D and beyond, J. Stat. Mech. 2003 (2020) 033106 [arXiv:1910.08459] [INSPIRE].
M.T. Tan and S. Ryu, Particle number fluctuations, Rényi entropy, and symmetry-resolved entanglement entropy in a two-dimensional Fermi gas from multidimensional bosonization, Phys. Rev. B 101 (2020) 235169 [arXiv:1911.01451] [INSPIRE].
S. Murciano, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement in two-dimensional systems via dimensional reduction, J. Stat. Mech. 2008 (2020) 083102 [arXiv:2003.11453] [INSPIRE].
L. Capizzi, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement entropy of excited states in a CFT, J. Stat. Mech. 2007 (2020) 073101 [arXiv:2003.04670] [INSPIRE].
X. Turkeshi, P. Ruggiero, V. Alba and P. Calabrese, Entanglement equipartition in critical random spin chains, Phys. Rev. B 102 (2020) 014455 [arXiv:2005.03331] [INSPIRE].
S. Murciano, G. Di Giulio and P. Calabrese, Entanglement and symmetry resolution in two dimensional free quantum field theories, JHEP 08 (2020) 073 [arXiv:2006.09069] [INSPIRE].
D.X. Horváth and P. Calabrese, Symmetry resolved entanglement in integrable field theories via form factor bootstrap, JHEP 11 (2020) 131 [arXiv:2008.08553] [INSPIRE].
D. Azses and E. Sela, Symmetry-resolved entanglement in symmetry-protected topological phases, Phys. Rev. B 102 (2020) 235157 [arXiv:2008.09332] [INSPIRE].
S. Zhao, C. Northe and R. Meyer, Symmetry-resolved entanglement in AdS3/CFT2 coupled to U(1) Chern-Simons theory, JHEP 07 (2021) 030 [arXiv:2012.11274] [INSPIRE].
D.X. Horváth, L. Capizzi and P. Calabrese, U(1) symmetry resolved entanglement in free 1 + 1 dimensional field theories via form factor bootstrap, JHEP 05 (2021) 197 [arXiv:2103.03197] [INSPIRE].
T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
R. Dijkgraaf, G.W. Moore, E.P. Verlinde and H.L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197 [hep-th/9608096] [INSPIRE].
R. Dijkgraaf, Fields, strings, matrices and symmetric products, hep-th/9912104 [INSPIRE].
P. Bantay, Symmetric products, permutation orbifolds and discrete torsion, Lett. Math. Phys. 63 (2003) 209 [hep-th/0004025] [INSPIRE].
C.A. Keller, Phase transitions in symmetric orbifold CFTs and universality, JHEP 03 (2011) 114 [arXiv:1101.4937] [INSPIRE].
J. de Boer, Large N elliptic genus and AdS/CFT correspondence, JHEP 05 (1999) 017 [hep-th/9812240] [INSPIRE].
R. Balian, Gain of information in a quantum measurement, Eur. J. Phys. 10 (1989) 208.
H.M. Wiseman and J.A. Vaccaro, Entanglement of Indistinguishable Particles Shared between Two Parties, Phys. Rev. Lett. 91 (2003) 097902 [quant-ph/0210002].
I. Klich and L.S. Levitov, Scaling of entanglement entropy and superselection rules, arXiv e-prints (2008) [arXiv:0812.0006].
A. Lukin et al., Probing entanglement in a many-body–localized system, Science 364 (2019) 256 [arXiv:1805.09819].
A. Belin, C.A. Keller and A. Maloney, Permutation Orbifolds in the large N Limit, arXiv:1509.01256 [INSPIRE].
A. Belin, C.A. Keller and I.G. Zadeh, Genus two partition functions and Rényi entropies of large c conformal field theories, J. Phys. A 50 (2017) 435401 [arXiv:1704.08250] [INSPIRE].
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Gerbershagen, M. Illuminating entanglement shadows of BTZ black holes by a generalized entanglement measure. J. High Energ. Phys. 2021, 187 (2021). https://doi.org/10.1007/JHEP10(2021)187
- AdS-CFT Correspondence
- Conformal Field Theory