Bit Threads and Holographic Monogamy

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


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.



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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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

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