Abstract
We compute the entanglement of purification (EoP) in a 2d free scalar field theory with various masses. This quantity measures correlations between two subsystems and is reduced to the entanglement entropy when the total system is pure. We obtain explicit numerical values by assuming minimal gaussian wave functionals for the purified states. We find that when the distance between the subsystems is large, the EoP behaves like the mutual information. However, when the distance is small, the EoP shows a characteristic behavior which qualitatively agrees with the conjectured holographic computation and which is different from that of the mutual information. We also study behaviors of mutual information in purified spaces and violations of monogamy/strong superadditivity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [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. 06 (2004) P06002 [hep-th/0405152] [INSPIRE].
H. Casini and M. Huerta, Entanglement and alpha entropies for a massive scalar field in two dimensions, J. Stat. Mech. 12 (2005) P12012 [cond-mat/0511014] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K., (2000).
R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865 [quant-ph/0702225] [INSPIRE].
I. Bengtsson and K. Zyczkowski, Geometry of quantum states, Cambridge University Press, Cambridge U.K., (2006).
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].
T. Takayanagi and K. Umemoto, Holographic entanglement of purification, Nature Phys. (2018) [arXiv:1708.09393] [INSPIRE].
P. Nguyen, T. Devakul, M.G. Halbasch, M.P. Zaletel and B. Swingle, Entanglement of purification: from spin chains to holography, JHEP 01 (2018) 098 [arXiv:1709.07424] [INSPIRE].
N. Bao and I.F. Halpern, Holographic inequalities and entanglement of purification, JHEP 03 (2018) 006 [arXiv:1710.07643] [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].
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].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
S. Bagchi and A.K. Pati, Monogamy, polygamy, and other properties of entanglement of purification, Phys. Rev. A 91 (2015) 042323 [arXiv:1502.01272].
J. Hauschild et al., Finding purifications with minimal entanglement, arXiv:1711.01288.
B.-B. Chen, L. Chen, Z. Chen, W. Li and A. Weichselbaum, Energy scales and exponential speedup in thermal tensor network simulations, arXiv:1801.00142.
C.H. Bennett, D.P. DiVincenzo, J.A. Smolin and W.K. Wootters, Mixed state entanglement and quantum error correction, Phys. Rev. A 54 (1996) 3824 [quant-ph/9604024] [INSPIRE].
V. Vedral, M.B. Plenio, M.A. Rippin and P.L. Knight, Quantifying entanglement, Phys. Rev. Lett. 78 (1997) 2275 [quant-ph/9702027] [INSPIRE].
R.R. Tucci, Entanglement of distillation and conditional mutual information, quant-ph/0202144.
M. Christandl and A. Winter, “Squashed entanglement”: an additive entanglement measure, J. Math. Phys. 45 (2004) 829 [quant-ph/0308088].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
P. Chaturvedi, V. Malvimat and G. Sengupta, Holographic quantum entanglement negativity, arXiv:1609.06609 [INSPIRE].
M.M. Wolf, G. Giedke, O. Krueger, R.F. Werner and J.I. Cirac, Gaussian entanglement of formation, Phys. Rev. A 69 (2004) 052320 [quant-ph/0306177].
Y. Huang, Computing quantum discord is NP-complete, New J. Phys. 16 (2014) 033027 [arXiv:1305.5941].
J. Chen and A. Winter, Non-additivity of the entanglement of purification (beyond reasonable doubt), arXiv:1206.1307.
M. Koashi and A. Winter, Monogamy of entanglement and other correlations, Phys. Rev. A 69 (2004) 022309 [quant-ph/0310037].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
G. Vidal, A class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [quant-ph/0610099].
G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165] [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].
M. Miyaji and T. Takayanagi, Surface/state correspondence as a generalized holography, PTEP 2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
N. Shiba, Entanglement entropy of two black holes and entanglement entropic force, Phys. Rev. D 83 (2011) 065002 [arXiv:1011.3760] [INSPIRE].
N. Shiba, Entanglement entropy of two spheres, JHEP 07 (2012) 100 [arXiv:1201.4865] [INSPIRE].
N. Shiba and T. Takayanagi, Volume law for the entanglement entropy in non-local QFTs, JHEP 02 (2014) 033 [arXiv:1311.1643] [INSPIRE].
V. Coffman, J. Kundu and W.K. Wootters, Distributed entanglement, Phys. Rev. A 61 (2000) 052306 [quant-ph/9907047] [INSPIRE].
M.F. Cornelio and M.C. de Oliveira, Strong superadditivity and monogamy of the Rényi measure of entanglement, Phys. Rev. A 81 (2010) 032332 [arXiv:0906.0332].
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].
P. Hayden, M. Headrick and A. Maloney, Holographic mutual information is monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [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: 1802.09545
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
Bhattacharyya, A., Takayanagi, T. & Umemoto, K. Entanglement of purification in free scalar field theories. J. High Energ. Phys. 2018, 132 (2018). https://doi.org/10.1007/JHEP04(2018)132
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2018)132