Abstract
Starting from the entanglement wedge of a multipartite mixed state we describe a purification procedure which involves the gluing of several copies. The resulting geometry has non-trivial topology and a single oriented boundary for each original boundary region. In the purified geometry the original multipartite entanglement wedge cross section is mapped to a minimal surface of a particular non-trivial homology class. In contrast, each original bipartite entanglement wedge cross section is mapped to the minimal wormhole throat around each boundary. Using the bit thread formalism we show how maximal flows for the bipartite and multipartite entanglement wedge cross section can be glued together to form maximal multiflows in the purified geometry. The defining feature differentiating the flows is given by the existence of threads which cross between different copies of the original entanglement wedge. Together these demonstrate a possible connection between multipartite entanglement and the topology of holographic spacetimes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys. 14 (2018) 573 [arXiv:1708.09393] [INSPIRE].
P. Nguyen, T. Devakul, M.G. Halbasch, M.P. Zaletel and B. Swingle, Entanglement of purification: from spin chains to holography, JHEP 01 (2018) 098 [arXiv:1709.07424] [INSPIRE].
N. Bao and I.F. Halpern, Holographic inequalities and entanglement of purification, JHEP 03 (2018) 006 [arXiv:1710.07643] [INSPIRE].
K. Umemoto and Y. Zhou, Entanglement of purification for multipartite states and its holographic dual, JHEP 10 (2018) 152 [arXiv:1805.02625] [INSPIRE].
N. Bao and I.F. Halpern, Conditional and multipartite entanglements of purification and holography, Phys. Rev. D 99 (2019) 046010 [arXiv:1805.00476] [INSPIRE].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys. 352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
M. Headrick and V.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav. 35 (2018) 10 [arXiv:1710.09516] [INSPIRE].
J. Harper and M. Headrick, Bit threads and holographic entanglement of purification, JHEP 08 (2019) 101 [arXiv:1906.05970] [INSPIRE].
D.-H. Du, C.-B. Chen and F.-W. Shu, Bit threads and holographic entanglement of purification, JHEP 08 (2019) 140 [arXiv:1904.06871] [INSPIRE].
N. Bao and N. Cheng, Multipartite reflected entropy, JHEP 10 (2019) 102 [arXiv:1909.03154] [INSPIRE].
J. Chu, R. Qi and Y. Zhou, Generalizations of reflected entropy and the holographic dual, JHEP 03 (2020) 151 [arXiv:1909.10456] [INSPIRE].
M. Headrick and V. Hubeny, Covariant bit threads, based on discussion, to appear.
M. Headrick and V. Hubeny, Covariant entanglement wedge cross section, based on discussion, to appear.
S.X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, Bit threads and holographic monogamy, Commun. Math. Phys. 376 (2019) 609 [arXiv:1808.05234] [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, arXiv:1905.00577 [INSPIRE].
M. Headrick and B. Zwiebach, Convex programs for minimal-area problems, Commun. Math. Phys. 377 (2020) 2217 [arXiv:1806.00449] [INSPIRE].
N. Engelhardt and A.C. Wall, Coarse graining holographic black holes, JHEP 05 (2019) 160 [arXiv:1806.01281] [INSPIRE].
D. Marolf, CFT sewing as the dual of AdS cut-and-paste, JHEP 02 (2020) 152 [arXiv:1909.09330] [INSPIRE].
K. Tamaoka, Entanglement wedge cross section from the dual density matrix, Phys. Rev. Lett. 122 (2019) 141601 [arXiv:1809.09109] [INSPIRE].
J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev. D 99 (2019) 106014 [arXiv:1808.00446] [INSPIRE].
N. Bao, G. Penington, J. Sorce and A.C. Wall, Beyond toy models: distilling tensor networks in full AdS/CFT, JHEP 11 (2019) 069 [arXiv:1812.01171] [INSPIRE].
C. Akers and P. Rath, Entanglement wedge cross sections require tripartite entanglement, JHEP 04 (2020) 208 [arXiv:1911.07852] [INSPIRE].
K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].
K. Krasnov, Black hole thermodynamics and Riemann surfaces, Class. Quant. Grav. 20 (2003) 2235 [gr-qc/0302073] [INSPIRE].
K. Skenderis and B.C. van Rees, Holography and wormholes in 2 + 1 dimensions, Commun. Math. Phys. 301 (2011) 583 [arXiv:0912.2090] [INSPIRE].
V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary wormholes and holographic entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].
N. Bao, Minimal purifications, wormhole geometries, and the complexity = action proposal, arXiv:1811.03113 [INSPIRE].
N. Bao, A. Chatwin-Davies and G.N. Remmen, Entanglement of purification and multiboundary wormhole geometries, JHEP 02 (2019) 110 [arXiv:1811.01983] [INSPIRE].
A. Bhattacharya, Multipartite purification, multiboundary wormholes, and islands in AdS3/CFT2, Phys. Rev. D 102 (2020) 046013 [arXiv:2003.11870] [INSPIRE].
N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully and M. Walter, The holographic entropy cone, JHEP 09 (2015) 130 [arXiv:1505.07839] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2006.02899
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Harper, J. Multipartite entanglement and topology in holography. J. High Energ. Phys. 2021, 116 (2021). https://doi.org/10.1007/JHEP03(2021)116
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2021)116