Bit Threads and Holographic Monogamy

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.

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Notes

  1. 1.

    MMI was also proven in the covariant setting in [43].

  2. 2.

    V. Hubeny has given a method to explicitly construct such a thread configuration, thereby establishing MMI, in certain cases [24].

  3. 3.

    Conversely, when additional structure is present, such as integer capacity edges in a graph, the statements that can be proven are often slightly stronger than what can be proven in the absence of such extra structure, e.g. by also obtaining results on the integrality of the flows.

  4. 4.

    \(\mathcal {M}\) may have an “internal” boundary \(\mathcal {B}\) that does not carry entropy, such as an orbifold fixed plane or end-of-the-world brane. This is accounted for in the Ryu–Takayanagi formula (2.1) by defining the homology to be relative to \(\mathcal {B}\), and in the max flow formula (2.4) by requiring the flow \(v^\mu \) to satisfy a Neumann boundary condition \(n_\mu v^\mu =0\) along \(\mathcal {B}\), and in the bit thread formula (2.4) by not allowing threads to end on \(\mathcal {B}\). See [22] for a fuller discussion. While we will not explicitly refer to internal boundaries in the rest of this paper, all of our results are valid in the presence of such a boundary.

  5. 5.

    The minimal surface is generically unique. In cases where it is not, we let m(A) denote any choice of minimal surface.

  6. 6.

    Flows are equivalent, via the Hodge star, to \((d-1)\)-calibrations [17]: the \((d-1)\)-form \(\omega =*(4G_\mathrm{N}g_{\mu \nu }v^\mu dx^\nu )\) is a calibration if and only if the vector field v is a flow. See [1] for further work using calibrations to compute holographic entanglement entropies.

  7. 7.

    It is conceptually natural to think of the threads as being microscopic but discrete, so that for example we can speak of the number of threads connecting two boundary regions. To be mathematically precise one could instead define a thread configuration as a continuous set \(\{\mathcal {C}\}\) of curves equipped with a measure \(\mu \). The density bound would then be imposed by requiring that, for every open subset s of \(\mathcal {M}\), \(\int d\mu \,\text {length}(\mathcal {C}\cap s)\le {{\,\mathrm{vol}\,}}(s)/4G_{\mathrm{N}}\), and the “number” of threads connecting two boundary regions would be defined as the total measure of that set of curves.

  8. 8.

    In addition to the threads connecting distinct boundary regions, there may be threads connecting a region to itself or simply forming a loop in the bulk. These will not play a role in our considerations.

  9. 9.

    One can alternatively work with the quantity \(I_3\), defined as the negative of \(-\,I_3\). However, when discussing holographic entanglement entropies, \(-\,I_3\) is more convenient since it is non-negative.

  10. 10.

    See [16, 22] for other proofs of the nesting property for flows.

  11. 11.

    While one may be tempted to similarly apply this theorem to n-party pure states for \(n>4\) to potentially prove other holographic inequalities, such efforts have not been successful to date (but see footnote 17 on p. 15).

  12. 12.

    We thank V. Hubeny for pointing this out to us. Futher details on this point will be presented elsewhere.

  13. 13.

    Strictly speaking, since a Bell pair has mutual information \(2\ln 2\), each thread represents \(1/\ln 2\) Bell pairs. If one really wanted each thread to represent one Bell pair, one could define the threads to have density \(|v|/\ln 2\), rather than |v|, for a given flow v.

  14. 14.

    As mentioned in footnote 4, the case where \(\mathcal {M}\) has an “internal boundary” \(\mathcal {B}\) is also physically relevant. In this case, \(\mathcal {B}\) not included in the decomposition into regions \(A_i\), all flows are required to satisfy the boundary condition \(n\cdot v=0\) on \(\mathcal {B}\), and homology relations are imposed relative to \(\mathcal {B}\). The reader can verify by following the proofs, with \(\partial \mathcal {M}\) replaced by \(\partial \mathcal {M}{\setminus }\mathcal {B}\), that all of our results hold in this case as well.

  15. 15.

    We refer the reader to [6] for an excellent guide to this rich subject, but we also recommend [22] for a short physicist-friendly introduction summarizing the concepts and results applied here.

  16. 16.

    Actually, a weaker condition is sufficient, namely, there being no flow from \(B_1\) to \(B_2\).

  17. 17.

    For example, one can prove the five-party cyclic inequality from [5] for networks by using similar techniques as presented in this paper by using a strengthened version of Theorem 3, known as the locking theorem (see e.g. [15]). Interestingly, the locking theorem does not appear to straightforwardly generalize to Riemannian geometries. (We thank V. Hubeny for pointing this out to us.)

  18. 18.

    We thank D. Marolf for useful discussions on this point.

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

    ADS  MathSciNet  Article  MATH  Google 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

    ADS  Article  MATH  Google 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

    ADS  MathSciNet  Article  MATH  Google 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

    ADS  MathSciNet  Article  Google 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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)

    Google 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

    ADS  MathSciNet  Article  Google 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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Du, D.H., Chen, C.B., Shu, F.W.: Bit threads and holographic entanglement of purification (2019)

  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)

    Article  Google Scholar 

  13. 13.

    Federer, H.: Real flat chains, cochains and variational problems. Indiana Univ. Math. J. 24, 351–407 (1974/1975)

  14. 14.

    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8(3), 399–404 (1956)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Frank, A., Karzanov, A.V., Sebo, A.: On integer multiflow maximization. SIAM J. Discrete Math. 10(1), 158–170 (1997)

    MathSciNet  Article  Google 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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Harvey, R., Lawson Jr., H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982). https://doi.org/10.1007/BF02392726

    MathSciNet  Article  MATH  Google 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

    ADS  Article  Google 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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Headrick, M.: General properties of holographic entanglement entropy. JHEP 03, 085 (2014). https://doi.org/10.1007/JHEP03(2014)085

    ADS  Article  Google Scholar 

  21. 21.

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

  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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Hernández Cuenca, S.: The Holographic Entropy Cone for Five Regions (2019)

  24. 24.

    Hubeny, V.E.: Bulk locality and cooperative flows. JHEP 12, 068 (2018). https://doi.org/10.1007/JHEP12(2018)068

    ADS  MathSciNet  Article  MATH  Google 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)

  26. 26.

    Kudler-Flam, J., Ryu, S.: Entanglement negativity and minimal entanglement wedge cross sections in holographic theories (2018)

  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

    MathSciNet  Article  Google 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

    MathSciNet  Article  MATH  Google 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

    MathSciNet  Article  MATH  Google 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)

  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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. 33.

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

    MathSciNet  MATH  Google 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

    ADS  MathSciNet  Article  MATH  Google 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

    ADS  MathSciNet  Article  MATH  Google 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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer, Berlin (2003)

    Google Scholar 

  38. 38.

    Strang, G.: Maximal flow through a domain. Math. Program. 26(2), 123–143 (1983). https://doi.org/10.1007/BF02592050

    MathSciNet  Article  MATH  Google 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

    Article  Google 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

    ADS  MathSciNet  Article  MATH  Google 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

    ADS  MathSciNet  Article  MATH  Google Scholar 

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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.

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Cui, S.X., Hayden, P., He, T. et al. Bit Threads and Holographic Monogamy. Commun. Math. Phys. 376, 609–648 (2020). https://doi.org/10.1007/s00220-019-03510-8

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