Abstract
The Ryu-Takayanagi prescription can be cast in terms of a set of microscopic threads that help visualize holographic entanglement in terms of distillation of EPR pairs. While this framework has been exploited for regions with a high degree of symmetry, we take the first steps towards understanding general entangling regions, focusing on AdS4. Inspired by simple constructions achieved for the case of disks and the half-plane, we reformulate bit threads in terms of a magnetic-like field generated by a current flowing through the boundary of the entangling region. The construction is possible for these highly symmetric settings, leading us to a modified Biot-Savart law in curved space that fully characterizes the entanglement structure of the state. For general entangling regions, the prescription breaks down as the corresponding modular Hamiltonians become inherently nonlocal. We develop a formalism for general shape deformations and derive a flow equation that accounts for these effects as a systematic expansion. We solve this equation for a complete set of small deformations and show that the structure of the expansion explicitly codifies the expected nonlocalities. Our findings are consistent with numerical results existing in the literature, and shed light on the fundamental nature of quantum entanglement as a nonlocal phenomenon.
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Acknowledgments
It is a pleasure to thank Aylon Pinto for his collaboration in the initial stages of this project. We also thank César Agón, Rafael Carrasco, Casey Cartwright, Veronika Hubeny, Andrew Svesko and Erik Tonni for useful discussions and comments on the manuscript. UG is supported by the Netherlands Organisation for Scientific Research (NWO) under the VICI grant VI.C.202.104. JFP is supported by the ‘Atracción de Talento’ program (2020-T1/TIC-20495, Comunidad de Madrid) and by the Spanish Research Agency (Agencia Estatal de Investigación) through the grants CEX2020-001007-S and PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by ERDF A way of making Europe.
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Gürsoy, U., Pedraza, J.F. & Planas, G.P. Holographic entanglement as nonlocal magnetism. J. High Energ. Phys. 2023, 91 (2023). https://doi.org/10.1007/JHEP09(2023)091
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DOI: https://doi.org/10.1007/JHEP09(2023)091