Advertisement

Bit Threads and Holographic Monogamy

  • Shawn X. Cui
  • Patrick Hayden
  • Temple HeEmail author
  • Matthew Headrick
  • Bogdan Stoica
  • Michael Walter
Article

Abstract

Bit threads provide an alternative description of holographic entanglement, replacing the Ryu–Takayanagi minimal surface with bulk curves connecting pairs of boundary points. We use bit threads to prove the monogamy of mutual information property of holographic entanglement entropies. This is accomplished using the concept of a so-called multicommodity flow, adapted from the network setting, and tools from the theory of convex optimization. Based on the bit thread picture, we conjecture a general ansatz for a holographic state, involving only bipartite and perfect-tensor type entanglement, for any decomposition of the boundary into four regions. We also give new proofs of analogous theorems on networks.

Notes

Acknowledgements

We would like to thank David Avis, Ning Bao, Veronika Hubeny, and Don Marolf for useful conversations. S.X.C. acknowledges the support from the Simons Foundation. P.H. and M.H. were supported by the Simons Foundation through the “It from Qubit” Simons Collaboration as well as, respectively, the Investigator and Fellowship programs. P.H. acknowledges additional support from CIFAR. P.H. and M.W. acknowledge support by AFOSR through Grant FA9550-16-1-0082. T.H. was supported in part by DOE Grant DE-FG02-91ER40654, and would like to thank Andy Strominger for his continued support and guidance. T.H. and B.S. would like to thank the Okinawa Institute for Science and Technology for their hospitality, where part of this work was completed. M.H. was also supported by the NSF under Career Award No. PHY-1053842 and by the U.S. Department of Energy under Grant DE-SC0009987. M.H. and B.S. would like to thank the MIT Center for Theoretical Physics for hospitality while this research was undertaken. M.H. and M.W. would also like to thank the Kavli Institute for Theoretical Physics, where this research was undertaken during the program “Quantum Physics of Information”; KITP is supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. The work of B.S. was supported in part by the Simons Foundation, and by the U.S. Department of Energy under Grant DE-SC-0009987. M.W. also acknowledges financial support by the NWO through Veni Grant No. 680-47-459. M.W. would also like to thank JILA for hospitality while this research was undertaken.

References

  1. 1.
    Bakhmatov, I., Deger, N.S., Gutowski, J., Colgain, E.O., Yavartanoo, H.: Calibrated entanglement entropy. JHEP 07, 117 (2017).  https://doi.org/10.1007/JHEP07(2017)117 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balasubramanian, V., Hayden, P., Maloney, A., Marolf, D., Ross, S.F.: Multiboundary wormholes and holographic entanglement. Class. Quantum Gravity 31, 185015 (2014).  https://doi.org/10.1088/0264-9381/31/18/185015 ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Bao, N., Halpern, I.F.: Holographic inequalities and entanglement of purification. JHEP 03, 006 (2018).  https://doi.org/10.1007/JHEP03(2018)006 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bao, N., Halpern, I.F.: Conditional and multipartite entanglements of purification and holography. Phys. Rev. D 99(4), 046010 (2019).  https://doi.org/10.1103/PhysRevD.99.046010 ADSCrossRefGoogle Scholar
  5. 5.
    Bao, N., Nezami, S., Ooguri, H., Stoica, B., Sully, J., Walter, M.: The holographic entropy cone. JHEP 09, 130 (2015).  https://doi.org/10.1007/JHEP09(2015)130 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    Casini, H., Huerta, M.: Remarks on the entanglement entropy for disconnected regions. JHEP 03, 048 (2009).  https://doi.org/10.1088/1126-6708/2009/03/048 ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chandrasekaran, R.: Multicommodity Maximum Flow Problems. https://www.utdallas.edu/~chandra/documents/networks/net7.pdf. Accessed 21 Feb 2018
  9. 9.
    Cherkassky, B.V.: A solution of a problem on multicommodity flows in a network. Ekonomika i matematicheski motody 13, 143–151 (1977)Google Scholar
  10. 10.
    Ding, D., Hayden, P., Walter, M.: Conditional mutual information of bipartite unitaries and scrambling. JHEP 12, 145 (2016).  https://doi.org/10.1007/JHEP12(2016)145 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Du, D.H., Chen, C.B., Shu, F.W.: Bit threads and holographic entanglement of purification (2019)Google Scholar
  12. 12.
    Elias, P., Feinstein, A., Shannon, C.E.: A note on the maximum flow through a network. IRE Trans Inf Theory 2(4), 117–119 (1956)CrossRefGoogle Scholar
  13. 13.
    Federer, H.: Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24, 351–407 (1974/1975)Google Scholar
  14. 14.
    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8(3), 399–404 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Frank, A., Karzanov, A.V., Sebo, A.: On integer multiflow maximization. SIAM J. Discrete Math. 10(1), 158–170 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Freedman, M., Headrick, M.: Bit threads and holographic entanglement. Commun. Math. Phys. 352(1), 407–438 (2017).  https://doi.org/10.1007/s00220-016-2796-3 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Harvey, R., Lawson Jr., H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982).  https://doi.org/10.1007/BF02392726 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hayden, P., Headrick, M., Maloney, A.: Holographic mutual information is monogamous. Phys. Rev. D 87(4), 046003 (2013).  https://doi.org/10.1103/PhysRevD.87.046003 ADSCrossRefGoogle Scholar
  19. 19.
    Hayden, P., Nezami, S., Qi, X.L., Thomas, N., Walter, M., Yang, Z.: Holographic duality from random tensor networks. JHEP 11, 009 (2016).  https://doi.org/10.1007/JHEP11(2016)009 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Headrick, M.: General properties of holographic entanglement entropy. JHEP 03, 085 (2014).  https://doi.org/10.1007/JHEP03(2014)085 ADSCrossRefGoogle Scholar
  21. 21.
    Headrick, M., Hubeny, V.E.: Covariant bit threads. (to appear) Google Scholar
  22. 22.
    Headrick, M., Hubeny, V.E.: Riemannian and Lorentzian flow-cut theorems. Class. Quantum Gravity 35(10), 105012 (2018).  https://doi.org/10.1088/1361-6382/aab83c ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hernández Cuenca, S.: The Holographic Entropy Cone for Five Regions (2019)Google Scholar
  24. 24.
    Hubeny, V.E.: Bulk locality and cooperative flows. JHEP 12, 068 (2018).  https://doi.org/10.1007/JHEP12(2018)068 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Karzanov, A., Lomonosov, M.V.: Systems of flows in undirected networks. In: Larichev, O.I. (ed.) Matematicheskoe Programmirovanie i dr. (Engl.: Mathematical Programming, and etc.), Issue 1, pp. 59–66. Inst. for System Studies (VNIISI) Press, Moscow (1978). (in Russian) Google Scholar
  26. 26.
    Kudler-Flam, J., Ryu, S.: Entanglement negativity and minimal entanglement wedge cross sections in holographic theories (2018)Google Scholar
  27. 27.
    Kupershtokh, V.L.: A generalization of the Ford–Fulkerson theorem to multipole networks. Cybernetics 7(3), 494–502 (1971).  https://doi.org/10.1007/BF01070459 MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lovász, L.: On some connectivity properties of Eulerian graphs. Acta Math. Acad. Sci. Hung. 28(1–2), 129–138 (1976).  https://doi.org/10.1007/BF01902503 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Maldacena, J., Susskind, L.: Cool horizons for entangled black holes. Fortschr. Phys. 61, 781–811 (2013).  https://doi.org/10.1002/prop.201300020 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Naves, G.: Notes on the Multicommodity Flow Problem. http://assert-false.net/callcc/Guyslain/Works/multiflows. Accessed 3 Oct 2017
  31. 31.
    Nezami, S., Walter, M.: Multipartite Entanglement in Stabilizer Tensor Networks (2016)Google Scholar
  32. 32.
    Nguyen, P., Devakul, T., Halbasch, M.G., Zaletel, M.P., Swingle, B.: Entanglement of purification: from spin chains to holography. JHEP 01, 098 (2018).  https://doi.org/10.1007/JHEP01(2018)098 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nozawa, R.: Max-flow min-cut theorem in an anisotropic network. Osaka J. Math. 27(4), 805–842 (1990)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Pastawski, F., Yoshida, B., Harlow, D., Preskill, J.: Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. JHEP 06, 149 (2015).  https://doi.org/10.1007/JHEP06(2015)149 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ryu, S., Takayanagi, T.: Aspects of holographic entanglement entropy. JHEP 08, 045 (2006).  https://doi.org/10.1088/1126-6708/2006/08/045 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ryu, S., Takayanagi, T.: Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602 (2006).  https://doi.org/10.1103/PhysRevLett.96.181602 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer, Berlin (2003)zbMATHGoogle Scholar
  38. 38.
    Strang, G.: Maximal flow through a domain. Math. Program. 26(2), 123–143 (1983).  https://doi.org/10.1007/BF02592050 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sullivan, J.M.: A crystalline approximation theorem for hypersurfaces. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), Princeton University. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9110403 (1990)
  40. 40.
    Umemoto, K., Takayanagi, T.: Entanglement of purification through holographic duality. Nat. Phys. 14(6), 573–577 (2018).  https://doi.org/10.1038/s41567-018-0075-2 CrossRefGoogle Scholar
  41. 41.
    Umemoto, K., Zhou, Y.: Entanglement of purification for multipartite states and its holographic dual. JHEP 10, 152 (2018).  https://doi.org/10.1007/JHEP10(2018)152 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Van Raamsdonk, M.: Building up spacetime with quantum entanglement. Gen. Relativ. Gravit. 42, 2323–2329 (2010).  https://doi.org/10.1007/s10714-010-1034-0,  https://doi.org/10.1142/S0218271810018529 [Int. J. Mod. Phys. D19,2429 (2010)]
  43. 43.
    Wall, A.C.: Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy. Class. Quantum Gravity 31(22), 225007 (2014).  https://doi.org/10.1088/0264-9381/31/22/225007 ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Stanford Institute for Theoretical PhysicsStanford UniversityStanfordUSA
  2. 2.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeUSA
  3. 3.Martin A. Fisher School of PhysicsBrandeis UniversityWalthamUSA
  4. 4.Center for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeUSA
  5. 5.Department of PhysicsBrown UniversityProvidenceUSA
  6. 6.Korteweg-de Vries Institute for Mathematics, Institute of Physics, Institute for Logic, Language and Computation, and QuSoftUniversity of AmsterdamAmsterdamThe Netherlands
  7. 7. Department of MathematicsVirginia TechBlacksburgUSA
  8. 8. Center for Quantum Mathematics and Physics (QMAP), Department of PhysicsUniversity of CaliforniaDavisUSA

Personalised recommendations