Abstract
In this work we study how entanglement of purification (EoP) and the new quantity of “complexity of purification” are related to each other using the EP = EW conjecture. First, we consider two strips in the same side of a boundary and study the relationships between the entanglement of purification of this mixed state and the parameters of the system such as dimension, temperature, length of the strips and the distance between them. Next, using the same setup, we introduce two definitions for the complexity of mixed states, complexity of purification (CoP) and the interval volume (VI). We study their connections to other parameters similar to the EoP case. Then, we extend our study to more general examples of BTZ black holes solution in massive gravity, charged black holes and multipartite systems. Finally, we give various interpretations of our results using resource theories such as LOCC and also bit thread picture.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE].
M. Alishahiha, Holographic Complexity, Phys. Rev.D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
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].
M. Ghodrati, Complexity growth in massive gravity theories, the effects of chirality and more, Phys. Rev. D 96 (2017) 106020 [arXiv:1708.07981] [INSPIRE].
M. Ghodrati, Complexity growth rate during phase transitions, Phys. Rev.D 98 (2018) 106011 [arXiv:1808.08164] [INSPIRE].
T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys.14 (2018) 573 [arXiv:1708.09393] [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].
P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP11 (2017) 097 [arXiv:1706.07056] [INSPIRE].
L.-P. Du, S.-F. Wu and H.-B. Zeng, Holographic complexity of the disk subregion in (2+1)-dimensional gapped systems, Phys. Rev. D 98 (2018) 066005 [arXiv:1803.08627] [INSPIRE].
C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, JHEP02 (2019) 145 [arXiv:1804.01561] [INSPIRE].
B. Chen, W.-M. Li, R.-Q. Yang, C.-Y. Zhang and S.-J. Zhang, Holographic subregion complexity under a thermal quench, JHEP07 (2018) 034 [arXiv:1803.06680] [INSPIRE].
Y. Ling, Y. Liu and C.-Y. Zhang, Holographic Subregion Complexity in Einstein-Born-Infeld theory, Eur. Phys. J.C 79 (2019) 194 [arXiv:1808.10169] [INSPIRE].
R.-Q. Yang, C.-Y. Zhang and W.-M. Li, Holographic entanglement of purification for thermofield double states and thermal quench, JHEP01 (2019) 114 [arXiv:1810.00420] [INSPIRE].
E. Cáceres, J. Couch, S. Eccles and W. Fischler, Holographic Purification Complexity, Phys. Rev.D 99 (2019) 086016 [arXiv:1811.10650] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev.D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
R. Abt, J. Erdmenger, M. Gerbershagen, C.M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, JHEP01 (2019) 012 [arXiv:1805.10298] [INSPIRE].
Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal, JHEP02 (2018) 072 [arXiv:1801.01137] [INSPIRE].
V.E. Hubeny, M. Rangamani and M. Rota, The holographic entropy arrangement, Fortsch. Phys.67 (2019) 1900011 [arXiv:1812.08133] [INSPIRE].
P.H.K. Horodecki, M. Horodecki and J. Oppenheim, Locking entanglement measures with a single qubit, Phys. Rev. Lett. 51 (2005) 200501 [quant-ph/0404096].
W.A. Christandl and A. Winter, Uncertainty, Monogamy, and Locking of Quantum Correlations, IEEE Trans. Inf. Theory51 (2005) 3159 [quant-ph/0501090].
Y.-T. Zhou, M. Ghodrati, X.-M. Kuang and J.-P. Wu, Evolutions of entanglement and complexity after a thermal quench in massive gravity theory, arXiv:1907.08453 [INSPIRE].
P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev.D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
C.G. Cevolani Lorenzo and S.-P. Laurent, Spreading of correlations in exactly solvable quantum models with long-range interactions in arbitrary dimensions, New J. Phys. 18 (2016) 093002 [arXiv:1604.05736].
E. Chitambar, Quantum correlations in high-dimensional states of high symmetry, Phys. Rev.A 86 (2012) 032110 [arXiv:1110.3057].
A. Bhattacharyya, T. Takayanagi and K. Umemoto, Entanglement of Purification in Free Scalar Field Theories, JHEP04 (2018) 132 [arXiv:1802.09545] [INSPIRE].
B.M. Terhal, M. Horodecki, D.W. Leung and D.P. DiVincenzo, The entanglement of purification, J. Math. Phys. 43 (2002) 4286 [quant-ph/0202044].
S.-Q. Lan, G.-Q. Li, J.-X. Mo and X.-B. Xu, A simple analysis of the mixed-state information metric in AdS 3/CF T 2, arXiv:1810.10441 [INSPIRE].
W.-Z. Guo, Entanglement of Purification and Projective Measurement in CFT, arXiv:1901.00330 [INSPIRE].
V.E. Hubeny, Bulk locality and cooperative flows, JHEP12 (2018) 068 [arXiv:1808.05313] [INSPIRE].
C.A. Agón, J. De Boer and J.F. Pedraza, Geometric Aspects of Holographic Bit Threads, JHEP05 (2019) 075 [arXiv:1811.08879] [INSPIRE].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys.352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
S.X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, Bit Threads and Holographic Monogamy, arXiv:1808.05234 [INSPIRE].
M. Alishahiha, K. Babaei Velni and M.R. Mohammadi Mozaffar, Black hole subregion action and complexity, Phys. Rev.D 99 (2019) 126016 [arXiv:1809.06031] [INSPIRE].
O. Ben-Ami and D. Carmi, On Volumes of Subregions in Holography and Complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].
R. Abt et al., Topological Complexity in AdS 3/CF T 2, Fortsch. Phys.66 (2018) 1800034 [arXiv:1710.01327] [INSPIRE].
P. Liu, Y. Ling, C. Niu and J.-P. Wu, Entanglement of Purification in Holographic Systems, arXiv:1902.02243 [INSPIRE].
K.X. Wei, C. Ramanathan and P. Cappellaro, Exploring localization in nuclear spin chains, Phys. Rev. Lett. 120 (2018) 070501 [arXiv:1612.05249].
L. Cevolani, J. Despres, G. Carleo, L. Tagliacozzo and L. Sanchez-Palencia, Universal scaling laws for correlation spreading in quantum systems with short- and long-range interactions, Phys. Rev.B 98 (2018) 024302 [arXiv:1706.00838] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
S.B. Giddings and M. Rota, Quantum information or entanglement transfer between subsystems, Phys. Rev.A 98 (2018) 062329 [arXiv:1710.00005] [INSPIRE].
J. Harper, M. Headrick and A. Rolph, Bit Threads in Higher Curvature Gravity, JHEP11 (2018) 168 [arXiv:1807.04294] [INSPIRE].
S.H. Hendi, B. Eslam Panah and S. Panahiyan, Massive charged BTZ black holes in asymptotically (a)dS spacetimes, JHEP05 (2016) 029 [arXiv:1604.00370] [INSPIRE].
M. Ghodrati and A. Naseh, Phase transitions in Bergshoeff-Hohm-Townsend massive gravity, Class. Quant. Grav. 34 (2017) 075009 [arXiv:1601.04403] [INSPIRE].
M. Ghodrati, K. Hajian and M.R. Setare, Revisiting Conserved Charges in Higher Curvature Gravitational Theories, Eur. Phys. J. C 76 (2016) 701 [arXiv:1606.04353] [INSPIRE].
M. Ghodrati, Beyond AdS Space-times, New Holographic Correspondences and Applications, Ph.D. Thesis, Michigan U., MCTP (2016) [arXiv:1609.04168] [INSPIRE].
M. Blake and D. Tong, Universal Resistivity from Holographic Massive Gravity, Phys. Rev.D 88 (2013) 106004 [arXiv:1308.4970] [INSPIRE].
Y. Huang, D.-J. Liu and X.-Z. Li, Superradiant instability of D-dimensional Reissner-Nordström-anti-de Sitter black hole mirror system, Int. J. Mod. Phys.D 26 (2017) 1750141 [arXiv:1606.00100] [INSPIRE].
R.A. Konoplya and A. Zhidenko, Stability of higher dimensional Reissner-Nordstrom-anti-de Sitter black holes, Phys. Rev.D 78 (2008) 104017 [arXiv:0809.2048] [INSPIRE].
S. Carlip, The (2+1)-Dimensional black hole, Class. Quant. Grav. 12 (1995) 2853 [gr-qc/9506079] [INSPIRE].
K. Umemoto and Y. Zhou, Entanglement of Purification for Multipartite States and its Holographic Dual, JHEP10 (2018) 152 [arXiv:1805.02625] [INSPIRE].
N. Bao and I.F. Halpern, Conditional and Multipartite Entanglements of Purification and Holography, Phys. Rev.D 99 (2019) 046010 [arXiv:1805.00476] [INSPIRE].
M. Headrick and V.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav. 35 (2018) 10 [arXiv:1710.09516] [INSPIRE].
A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev.D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
M. Navascués, Pure state estimation and the characterization of entanglement, Phys. Rev. Lett.100 (2008) 070503 [arXiv:0707.4398].
S. Bandyopadhyay, S. Halder and M. Nathanson, Optimal resource states for local state discrimination, Phys. Rev.A 97 (2018) 022314 [arXiv:1709.10302].
F. Verstraete and H. Verschelde, Optimal teleportation with a mixed state of two qubits, Phys. Rev. Lett.90 (2003) 097901 [quant-ph/0303007].
K.P. Seshadreesan and M.M. Wilde, Fidelity of recovery, squashed entanglement and measurement recoverability, Phys. Rev. A 92 (2015) 042321 [arXiv:1410.1441].
S.-M.F. Ming Li and X. Li-Jost, Quantum entanglement: Separability, measure, fidelity of teleportation, and distillation, Adv. Math. Phys.2010 (2010) 301072 [arXiv:1012.4706].
K. Surmacz, J. Nunn, F.C. Waldermann, Z. Wang, I.A. Walmsley and D. Jaksch, Entanglement fidelity of quantum memories, Phys. Rev.A 74 (2006) 050302 [quant-ph/0608098].
Y. Guo, Strict entanglement monotonicity under local operations and classical communication, Phys. Rev.A 99 (2019) 022338 [arXiv:1904.01183].
D.-H. Du, C.-B. Chen and F.-W. Shu, Bit threads and holographic entanglement of purification, arXiv:1904.06871 [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].
M. Miyaji and T. Takayanagi, Surface/State Correspondence as a Generalized Holography, PTEP2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
P. Caputa, M. Miyaji, T. Takayanagi and K. Umemoto, Holographic Entanglement of Purification from Conformal Field Theories, Phys. Rev. Lett. 122 (2019) 111601 [arXiv:1812.05268] [INSPIRE].
A.R. Brown, H. Gharibyan, H.W. Lin, L. Susskind, L. Thorlacius and Y. Zhao, Complexity of Jackiw-Teitelboim gravity, Phys. Rev. D 99 (2019) 046016 [arXiv:1810.08741] [INSPIRE].
G. Evenbly and G. Vidal, Tensor network renormalization yields the multiscale entanglement renormalization ansatz, Phys. Rev. Lett. 115 (2015) 200401 [arXiv:1502.05385].
B. Czech, L. Lamprou, S. Mccandlish and J. Sully, Modular Berry Connection for Entangled Subregions in AdS/CFT, Phys. Rev. Lett.120 (2018) 091601 [arXiv:1712.07123] [INSPIRE].
M. Ghodrati, Entanglement of Purification and Modular Berry Flow, to appear.
V. Balasubramanian, M. DeCross, A. Kar and O. Parrikar, Binding Complexity and Multiparty Entanglement, JHEP02 (2019) 069 [arXiv:1811.04085] [INSPIRE].
M. Ghodrati, Schwinger Effect and Entanglement Entropy in Confining Geometries, Phys. Rev. D 92 (2015) 065015 [arXiv:1506.08557] [INSPIRE].
M. Ghodrati, Hyperscaling Violating Solution in Coupled Dilaton-Squared Curvature Gravity, Phys. Rev.D 90 (2014) 044055 [arXiv:1404.5399] [INSPIRE].
M. Ghodrati, X.-M. Kuang, B. Wang, C.-Y. Zhang and Y.-T. Zhou, The connection between holographic entanglement and complexity of purification, arXiv:1902.02475 [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: 1902.02475
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ghodrati, M., Kuang, XM., Wang, B. et al. The connection between holographic entanglement and complexity of purification. J. High Energ. Phys. 2019, 9 (2019). https://doi.org/10.1007/JHEP09(2019)009
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2019)009