Abstract
It was recently conjectured that the quantum complexity of a holographic boundary state can be computed by evaluating the gravitational action on a bulk region known as the Wheeler-DeWitt patch. We apply this complexity=action duality to evaluate the ‘complexity of formation’ [1, 2], i.e. the additional complexity arising in preparing the entangled thermofield double state with two copies of the boundary CFT compared to preparing the individual vacuum states of the two copies. We find that for boundary dimensions d > 2, the difference in the complexities grows linearly with the thermal entropy at high temperatures. For the special case d = 2, the complexity of formation is a fixed constant, independent of the temperature. We compare these results to those found using the complexity=volume duality.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic complexity equals bulk action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [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].
J. Watrous, Quantum computational complexity, in Encyclopedia of complexity and systems science, R.A. Meyers ed., Springer, Germany (2009), arXiv:0804.3401.
T.J. Osborne, Hamiltonian complexity, Rept. Prog. Phys. 75 (2012) 022001 [arXiv:1106.5875].
S. Gharibian, Y. Huang, Z. Landau and S.W. Shin, Quantum hamiltonian complexity, Foundations and Trends in Theoretical Computer Science volume 10, NOW, Boston U.S.A. (2015), arXiv:1401.3916.
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678].
L. Susskind and Y. Zhao, Switchbacks and the bridge to nowhere, arXiv:1408.2823 [INSPIRE].
M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum computation as geometry, Science 311 (2006) 1133 [quant-ph/0603161].
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, Quant. Inf. Comput. 8 (2008) 861 [quant-ph/0701004].
D.A. Roberts and B. Yoshida, Chaos and complexity by design, arXiv:1610.04903 [INSPIRE].
L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, arXiv:1609.00207.
J.M. Maldacena, Eternal black holes in Anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
R.C. Myers, Stress tensors and Casimir energies in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 046002 [hep-th/9903203] [INSPIRE].
R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].
T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero energy states, JHEP 06 (1999) 036 [hep-th/9906040] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577.
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
Y.M. Cho and I.P. Neupane, Anti-de Sitter black holes, thermal phase transition an holography in higher curvature gravity, Phys. Rev. D 66 (2002) 024044 [hep-th/0202140].
D. Carmi, R.C. Myers and P. Rath, omments on Holographic Complexity, arXiv:1612.00433 [INSPIRE].
J.W. York, Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
K. Parattu, S. Chakraborty, B.R. Majhi and T. Padmanabhan, A boundary term for the gravitational action with null boundaries, Gen. Rel. Grav. 48 (2016) 94 [arXiv:1501.01053] [INSPIRE].
G. Hayward, Gravitational action for space-times with nonsmooth boundaries, Phys. Rev. D 47 (1993) 3275 [INSPIRE].
D. Brill and G. Hayward, Is the gravitational action additive?, Phys. Rev. D 50 (1994) 4914 [gr-qc/9403018] [INSPIRE].
D. Carmi et al., On the time dependence of holographic complexity, in preparation.
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
C. Fefferman and C.R. Graham, Conformal invariants, in Elie Cartan et les Mathématiques d’aujourd’hui, Astérisque (1985) 95.
C. Fefferman and C.R. Graham, The ambient metric, arXiv:0710.0919 [INSPIRE].
A. Buchel et al., Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [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.
O. Coussaert and M. Henneaux, Supersymmetry of the (2 + 1) black holes, Phys. Rev. Lett. 72 (1994) 183 [hep-th/9310194].
J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume and complexity, arXiv:1610.02038.
M. Alishahiha, Holographic complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
O. Ben-Ami and D. Carmi, On volumes of subregions in holography and complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].
S. Lloyd, Ultimate physical limits to computation, Nature 406 (2000) 1047 [quant-ph/9908043].
R.-Q. Yang, Strong energy condition and the fastest computers, arXiv:1610.05090 [INSPIRE].
G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [INSPIRE].
G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [quant-ph/0610099].
G. Vidal, Entanglement renormalization: an introduction, in Understanding quantum phase transitions, L.D. Carr ed., Taylor & Francis, Boca Raton, U.S.A. (2010), arXiv:0912.1651.
G. Evenbly and G. Vidal, Tensor network renormalization yields the multi-scale entanglement renormalization ansatz, Phys. Rev. Lett. 115 (2015) 200401 [arXiv:1502.05385].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral geometry and holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
B. Czech et al., Tensor network quotient takes the vacuum to the thermal state, Phys. Rev. B 94 (2016) 085101 [arXiv:1510.07637] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Tensor networks from kinematic space, JHEP 07 (2016) 100 [arXiv:1512.01548] [INSPIRE].
R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang and R.-H. Peng, Action growth for AdS black holes, JHEP 09 (2016) 161 [arXiv:1606.08307] [INSPIRE].
D. Roberts, private communication.
A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60 (1999) 064018 [hep-th/9902170] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
G. Evenbly and G. Vidal, Tensor network renormalization, Phys. Rev. Lett. 115 (2015) 180405 [arXiv:1412.0732].
R. Orus, A practical introduction to tensor networks: matrix product states and projected entangled pair states, Annals Phys. 349 (2014) 117 [arXiv:1306.2164] [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].
D. Marolf, H. Maxfield, A. Peach and S.F. Ross, Hot multiboundary wormholes from bipartite entanglement, Class. Quant. Grav. 32 (2015) 215006 [arXiv:1506.04128] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1610.08063
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Chapman, S., Marrochio, H. & Myers, R.C. Complexity of formation in holography. J. High Energ. Phys. 2017, 62 (2017). https://doi.org/10.1007/JHEP01(2017)062
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2017)062