Bit threads and holographic entanglement of purification

  • Jonathan HarperEmail author
  • Matthew Headrick
Open Access
Regular Article - Theoretical Physics


Generalizing the bit thread formalism, we use convex duality to derive dual flow programs to the bipartite and multipartite holographic entanglement of purification proposals and then prove several inequalities using these constructions. In the multipartite case we find the flows exhibit novel behavior which allows for a constrained flux on the boundary of the homology region. We show this flux can be made distinct from bi-partite terms and reflects the truly multipartite portion of the holographic entanglement of purification.


AdS-CFT Correspondence Differential and Algebraic Geometry 


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


  1. [1]
    T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys.14 (2018) 573 [arXiv:1708.09393] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    P. Nguyen, T. Devakul, M.G. Halbasch, M.P. Zaletel and B. Swingle, Entanglement of purification: from spin chains to holography, JHEP01 (2018) 098 [arXiv:1709.07424] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, arXiv:1905.00577 [INSPIRE].
  4. [4]
    J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev.D 99 (2019) 106014 [arXiv:1808.00446] [INSPIRE].ADSMathSciNetGoogle Scholar
  5. [5]
    K. Tamaoka, Entanglement Wedge Cross Section from the Dual Density Matrix, Phys. Rev. Lett. 122 (2019) 141601 [arXiv:1809.09109] [INSPIRE].
  6. [6]
    M. Miyaji and T. Takayanagi, Surface/State Correspondence as a Generalized Holography, PTEP2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  7. [7]
    N. Bao, Minimal Purifications, Wormhole Geometries and the Complexity=Action Proposal, arXiv:1811.03113 [INSPIRE].
  8. [8]
    W.-Z. Guo, Entanglement of Purification and Projective Measurement in CFT, arXiv:1901.00330 [INSPIRE].
  9. [9]
    N. Bao, A. Chatwin-Davies, J. Pollack and G.N. Remmen, Towards a Bit Threads Derivation of Holographic Entanglement of Purification, JHEP07 (2019) 152 [arXiv:1905.04317] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  10. [10]
    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].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    A. Bhattacharyya, T. Takayanagi and K. Umemoto, Entanglement of Purification in Free Scalar Field Theories, JHEP04 (2018) 132 [arXiv:1802.09545] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    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].
  13. [13]
    A. Bhattacharyya, A. Jahn, T. Takayanagi and K. Umemoto, Entanglement of Purification in Many Body Systems and Symmetry Breaking, Phys. Rev. Lett.122 (2019) 201601 [arXiv:1902.02369] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    S. Bagchi, Monogamy, polygamy, and other properties of entanglement of purification, Phys. Rev.A 91 (2015) 042323.ADSCrossRefGoogle Scholar
  15. [15]
    P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev.D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].ADSGoogle Scholar
  16. [16]
    M. Headrick, General properties of holographic entanglement entropy, JHEP03 (2014) 085 [arXiv:1312.6717] [INSPIRE].
  17. [17]
    C.A. Agón, J. De Boer and J.F. Pedraza, Geometric Aspects of Holographic Bit Threads, JHEP05 (2019) 075 [arXiv:1811.08879] [INSPIRE].
  18. [18]
    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].
  19. [19]
    J. Kudler-Flam, I. MacCormack and S. Ryu, Holographic entanglement contour, bit threads and the entanglement tsunami, J. Phys.A 52 (2019) 325401 [arXiv:1902.04654] [INSPIRE].MathSciNetGoogle Scholar
  20. [20]
    K. Umemoto and Y. Zhou, Entanglement of Purification for Multipartite States and its Holographic Dual, JHEP10 (2018) 152 [arXiv:1805.02625] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    N. Bao and I.F. Halpern, Conditional and Multipartite Entanglements of Purification and Holography, Phys. Rev.D 99 (2019) 046010 [arXiv:1805.00476] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, (2004), [].
  23. [23]
    M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys.352 (2017) 407 [arXiv:1604.00354] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    M. Headrick and V.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav.35 (2018) 10 [arXiv:1710.09516] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  25. [25]
    D.-H. Du, C.-B. Chen and F.-W. Shu, Bit threads and holographic entanglement of purification, arXiv:1904.06871 [INSPIRE].
  26. [26]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    N. Bao and I.F. Halpern, Holographic Inequalities and Entanglement of Purification, JHEP03 (2018) 006 [arXiv:1710.07643] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    S.X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, Bit Threads and Holographic Monogamy, arXiv:1808.05234 [INSPIRE].
  29. [29]
    N. Bao, A. Chatwin-Davies and G.N. Remmen, Entanglement of Purification and Multiboundary Wormhole Geometries, JHEP02 (2019) 110 [arXiv:1811.01983] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    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].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    M. Headrick and V. Hubeny, Covariant bit threads, to appear.Google Scholar
  33. [33]
    M. Christandl and A. Winter, “squashed entanglement” — an additive entanglement measure, J. Math. Phys. 45 (2004) 829 [quant-ph/0308088].
  34. [34]
    R.R. Tucci, Entanglement of distillation and conditional mutual information, quant-ph/0202144.

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Martin Fisher School of PhysicsBrandeis UniversityWalthamU.S.A.

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