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Bit Threads and Holographic Entanglement

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Abstract

The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, we invoke the notion of a “flow”, defined as a divergenceless norm-bounded vector field, or equivalently a set of Planck-thickness “bit threads”. The entanglement entropy of a boundary region is given by the maximum flux out of it of any flow, or equivalently the maximum number of bit threads that can emanate from it. The threads thus represent entanglement between points on the boundary, and naturally implement the holographic principle. As we explain, this new picture clarifies several conceptual puzzles surrounding the RT formula. We give flow-based proofs of strong subadditivity and related properties; unlike the ones based on minimal surfaces, these proofs correspond in a transparent manner to the properties’ information-theoretic meanings. We also briefly discuss certain technical advantages that the flows offer over minimal surfaces. In a mathematical appendix, we review the max flow-min cut theorem on networks and on Riemannian manifolds, and prove in the network case that the set of max flows varies Lipshitz continuously in the network parameters.

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References

  1. Ryu S., Takayanagi T.: Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602 (2006) ar**v:hep-th/0603001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Ryu S., Takayanagi T.: Aspects of holographic entanglement entropy. JHEP 08, 045 (2006) ar**v:hep-th/0605073

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Hubeny V.E., Rangamani M., Takayanagi T.: A Covariant holographic entanglement entropy proposal. JHEP 07, 062 (2007) ar**v:0705.0016

    Article  ADS  MathSciNet  Google Scholar 

  4. Hirata T., Takayanagi T.: AdS/CFT and strong subadditivity of entanglement entropy. JHEP 02, 042 (2007) ar**v:hep-th/0608213

    Article  ADS  MathSciNet  Google Scholar 

  5. Nishioka, T., Takayanagi, T.: AdS bubbles, entropy and closed string tachyons. JHEP 01, 090 (2007). ar**v:hep-th/0611035

  6. Klebanov, I.R., Kutasov, D., Murugan, A.: Entanglement as a probe of confinement. Nucl. Phys. B 796, 274–293 (2008). ar**v:0709.2140

  7. Headrick, M.: Entanglement Renyi entropies in holographic theories. Phys. Rev. D 82, 126010 (2010). ar**v:1006.0047

  8. Headrick, M.: General properties of holographic entanglement entropy. JHEP 03, 085 (2014). ar**v:1312.6717

  9. Headrick, M., Takayanagi, T.: A Holographic proof of the strong subadditivity of entanglement entropy. Phys. Rev. D 76, 106013 (2007). ar**v:0704.3719

  10. Federer H.: Real flat chains, cochains and variational problems. Indiana Univ. Math J. 24, 351–407 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Strang G.: Maximal flow through a domain. Math. Program. 26(2), 123–143 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nozawa R.: Max-flow min-cut theorem in an anisotropic network. Osaka J. Math. 27(4), 805–842 (1990)

    MathSciNet  MATH  Google Scholar 

  13. Pastawski, F., Yoshida, B., Harlow, D., Preskill, J.: Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. JHEP 06, 149 (2015). ar**v:1503.06237

  14. ’t Hooft, G.: Dimensional reduction in quantum gravity. In: Salamfest, pp. 0284–296 (1993). ar**v:gr-qc/9310026

  15. Susskind, L.: The World as a hologram. J. Math. Phys. 36, 6377–6396 (1995). ar**v:hep-th/9409089

  16. Headrick, M., Hubeny, V.E., Lawrence, A., Rangamani, M.: Causality and holographic entanglement entropy. JHEP 12, 162 (2014). ar**v:1408.6300

  17. Czech B., Karczmarek J.L., Nogueira F., Van Raamsdonk M.: The Gravity dual of a density matrix. Class. Quant. Grav. 29, 155009 (2012) ar**v:1204.1330

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Jafferis, D.L., Lewkowycz, A., Maldacena, J., Suh, S.J.: Relative entropy equals bulk relative entropy. JHEP 06, 004 (2016). ar**v:1512.06431

  19. Headrick, M., Hubeny, V.: Covariant holographic bit threads (to appear)

  20. Hayden P., Headrick M., Maloney A.: Holographic mutual information is monogamous. Phys. Rev. D 87(4), 046003 (2013) ar**v:1107.2940

    Article  ADS  Google Scholar 

  21. Bao, N., Nezami, S., Ooguri, H., Stoica, B., Sully, J., Walter, M.: The holographic entropy cone. JHEP 09, 130 (2015). ar**v:1505.07839

  22. Balasubramanian V., Hayden P., Maloney A., Marolf D., Ross S.F.: Multiboundary Wormholes and Holographic Entanglement. Class. Quant. Grav. 31, 185015 (2014) ar**v:1406.2663

    Article  ADS  MATH  Google Scholar 

  23. Cui S.X., Freedman M.H., Sattath O., Stong R., Minton G.: Quantum max-flow/min-cut. J. Math. Phys. 57, 062206 (2016) ar**v:1508.04644

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Hastings, M.: The asymptotics of quantum max-flow min-cut. ar**v:1603.03717

  25. Weyl H.: Über die asymptotische Verteilung der Eigenwerte. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen 2, 110–117 (1911)

    MATH  Google Scholar 

  26. Jakobson D., Polterovich I.: Estimates from below for the spectral function and for the remainder in local Weyl’s law. Geom. Funct. Anal. 17(3), 806–838 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hung L.-Y., Myers R.C., Smolkin M.: On holographic entanglement entropy and higher curvature gravity. JHEP 04, 025 (2011)

    Article  ADS  Google Scholar 

  28. de Boer J., Kulaxizi M., Parnachev A.: Holographic entanglement entropy in Lovelock gravities. JHEP 07, 109 (2011) ar**v:1101.5781

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Dong X.: Holographic entanglement entropy for general higher derivative gravity. JHEP 01, 044 (2014) ar**v:1310.5713

    Article  ADS  MATH  Google Scholar 

  30. de Boer J., Jottar J.I.: Entanglement entropy and higher spin holography in AdS3. JHEP 04, 089 (2014) ar**v:1306.4347

    Article  Google Scholar 

  31. Ammon M., Castro A., Iqbal N.: Wilson lines and entanglement entropy in higher spin gravity. JHEP 10, 110 (2013) ar**v:1306.4338

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Castro A., Llabres E.: Unravelling holographic entanglement entropy in higher spin theories. JHEP 03, 124 (2015) ar**v:1410.2870

    Article  MathSciNet  Google Scholar 

  33. Faulkner T., Lewkowycz A., Maldacena J.: Quantum corrections to holographic entanglement entropy. JHEP 11, 074 (2013) ar**v:1307.2892

    Article  ADS  Google Scholar 

  34. Maldacena J., Susskind L.: Cool horizons for entangled black holes. Fortsch. Phys. 61, 781–811 (2013) ar**v:1306.0533

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Balasubramanian V., Chowdhury B.D., Czech B., de Boer J.: Entwinement and the emergence of spacetime. JHEP 01, 048 (2015) ar**v:1406.5859

    Article  ADS  Google Scholar 

  36. Czech B., Lamprou L.: Holographic definition of points and distances. Phys. Rev. D 90, 106005 (2014) ar**v:1409.4473

    Article  ADS  Google Scholar 

  37. Czech B., Lamprou L., McCandlish S., Sully J.: Integral Geometry and Holography. JHEP 10, 175 (2015) ar**v:1505.05515

    Article  ADS  MathSciNet  Google Scholar 

  38. Engelhardt N., Wall A.C.: Extremal Surface Barriers. JHEP 03, 068 (2014) ar**v:1312.3699

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Freivogel B., Jefferson R.A., Kabir L., Mosk B., Yang I.-S.: Casting Shadows on Holographic Reconstruction. Phys. Rev. D 91(8), 086013 (2015) ar**v:1412.5175

    Article  ADS  Google Scholar 

  40. Ford L.R. Jr., Fulkerson D.R.: Maximal flow through a network. Can. J. Math. 8, 399–404 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  41. Elias P., Feinstein A., Shannon C.: Note on maximal flow through a network. IRE Trans. Inf. Theory IT-2, 117–199 (1956)

    Article  Google Scholar 

  42. Harvey R., Lawson H.B. Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  43. Federer H.: Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)

    Google Scholar 

  44. Wikipedia, Calibrated geometry—wikipedia, the free encyclopedia (2015). Online accessed 11 Jan 2016

  45. Young L.C.: Some extremal questions for simplicial complexes. V. The relative area of a Klein bottle. Rend. Circ. Mat. Palermo (2) 12, 257–274 (1963)

    Article  MathSciNet  Google Scholar 

  46. White B.: The least area bounded by multiples of a curve. Proc. Am. Math. Soc. 90(2), 230–232 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  47. Morgan F.: Area-minimizing currents bounded by higher multiples of curves. Rend. Circ. Mat. Palermo (2) 33(1), 37–46 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hardt R., Simon L.: Area minimizing hypersurfaces with isolated singularities. J. Reine Angew. Math. 362, 102–129 (1985)

    MathSciNet  MATH  Google Scholar 

  49. Bombieri E., De Giorgi E., Giusti E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Simons J.: Minimal cones, Plateau’s problem, and the Bernstein conjecture. Proc. Natl. Acad. Sci. USA 58, 410–411 (1967)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Chodosh, O.: Co-dimension one minimizing verifolds. MathOverflow. http://mathoverflow.net/q/181459 (version: 2014-09-21)

  52. Smale N.: Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional manifolds. Commun. Anal. Geom. 1(2), 217–228 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  53. Morgan F.: Geometric Measure Theory: A Beginner’s Guide. Elsevier/Academic Press, Amsterdam (2016)

    Book  MATH  Google Scholar 

  54. Zhang, Y.: On extending calibrations. ar**v:1501.06163

  55. Sullivan, J.M.: A crystalline approximation theorem for hypersurfaces. ProQuest LLC, Ann Arbor, Thesis (Ph.D.), Princeton University (1990)

  56. Menger K.: Über reguläre Baumkurven. Math. Ann. 96(1), 572–582 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  57. Wikipedia, Ford–Fulkerson algorithm—Wikipedia, the free encyclopedia (2015). Online accessed 11 Jan 2016

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Authors and Affiliations

  1. Station Q, Microsoft Research, Santa Barbara, CA, 93106, USA

    Michael Freedman

  2. Martin Fisher School of Physics, Brandeis University, Waltham, MA, 02453, USA

    Matthew Headrick

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  1. Michael Freedman
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Correspondence to Matthew Headrick.

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Communicated by X. Yin

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Freedman, M., Headrick, M. Bit Threads and Holographic Entanglement. Commun. Math. Phys. 352, 407–438 (2017). https://doi.org/10.1007/s00220-016-2796-3

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