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Bit Threads and Holographic Monogamy

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

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

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

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

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

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

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

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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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Author notes

  1. Shawn X. Cui

    Present address: Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

  2. Temple He

    Present address: Center for Quantum Mathematics and Physics (QMAP), Department of Physics, University of California, Davis, CA, 95616, USA

Authors and Affiliations

  1. Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305, USA

    Shawn X. Cui & Patrick Hayden

  2. Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA, 02138, USA

    Temple He

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

    Matthew Headrick & Bogdan Stoica

  4. Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Matthew Headrick

  5. Department of Physics, Brown University, Providence, RI, 02912, USA

    Bogdan Stoica

  6. Korteweg-de Vries Institute for Mathematics, Institute of Physics, Institute for Logic, Language and Computation, and QuSoft, University of Amsterdam, 1098 XG, Amsterdam, The Netherlands

    Michael Walter

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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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