Abstract
We derive several new reformulations of the Hubeny-Rangamani-Takayanagi covariant holographic entanglement entropy formula. These include: (1) a minimax formula, which involves finding a maximal-area achronal surface on a timelike hypersurface homologous to D(A) (the boundary causal domain of the region A whose entropy we are calculating) and minimizing over the hypersurface; (2) a max V-flow formula, in which we maximize the flux through D(A) of a divergenceless bulk 1-form V subject to an upper bound on its norm that is non-local in time; and (3) a min U-flow formula, in which we minimize the flux over a bulk Cauchy slice of a divergenceless timelike 1-form U subject to a lower bound on its norm that is non-local in space. The two flow formulas define convex programs and are related to each other by Lagrange duality. For each program, the optimal configurations dynamically find the HRT surface and the entanglement wedges of A and its complement. The V-flow formula is the covariant version of the Freedman-Headrick bit thread reformulation of the Ryu-Takayanagi formula. We also introduce a measure-theoretic concept of a “thread distribution”, and explain how Riemannian flows, V-flows, and U-flows can be expressed in terms of thread distributions.
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Acknowledgments
We would like to thank the following individuals for helpful conversations and feedback on the manuscript: C. Agon, N. Engelhardt, M. Freedman, G. Grimaldi, J. Harper, J. Pedraza, M. Rangamani, A. Rolph, M. Rota, B. Stoica, H. Verlinde, E. Witten.
We would also like to thank UC Davis, the Simons Center for Geometry and Physics, the Aspen Center for Physics, the Kavli Institute for Theoretical Physics, the Perimeter Institute for Theoretical Physics, the Galileo Galilei Institute, and the MIT Center for Theoretical Physics for hospitality while this research was being carried out.
MH is supported in part by the Simons Foundation through the It from Qubit Collaboration and by the U.S. Department of Energy grant DE-SC0009986. VH has been supported in part by the U.S. Department of Energy grant DE-SC0009999. Both authors are also supported by the U.S. Department of Energy grant DE-SC0020360 under the HEP-QIS QuantISED program.
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Headrick, M., Hubeny, V.E. Covariant bit threads. J. High Energ. Phys. 2023, 180 (2023). https://doi.org/10.1007/JHEP07(2023)180
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DOI: https://doi.org/10.1007/JHEP07(2023)180