Abstract
We generalize bit threads to hyperthreads in the context of holographic spacetimes. We define a “k-thread” to be a hyperthread which connects k different boundary regions and posit that it may be considered as a unit of k-party entanglement. Using this new object, we show that the contribution of hyperthreads to calculations of holographic entanglement entropy are generically finite. This is accomplished by constructing a surface whose area determines their maximum allowed contribution. We also identify surfaces whose area is proportional to the maximum number of k-threads, motivating a possible measure of multipartite entanglement. We use this to make connections to the current understanding of multipartite entanglement in holographic spacetimes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Boyd, Convex optimization, Cambridge University Press (2004) [DOI].
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) 105012.
N. Bao and J. Harper, Bit threads on hypergraphs, arXiv:2012.07872 [INSPIRE].
M. Headrick and V. Hubeny, Covariant bit threads, 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].
N. Bao, N. Cheng, S. Hernández-Cuenca and V.P. Su, The Quantum Entropy Cone of Hypergraphs, SciPost Phys. 9 (2020) 5 [arXiv:2002.05317] [INSPIRE].
J. Harper and M. Headrick, Bit threads and holographic entanglement of purification, JHEP 08 (2019) 101 [arXiv:1906.05970] [INSPIRE].
C. Akers and P. Rath, Entanglement Wedge Cross Sections Require Tripartite Entanglement, JHEP 04 (2020) 208 [arXiv:1911.07852] [INSPIRE].
P. Hayden, O. Parrikar and J. Sorce, The Markov gap for geometric reflected entropy, arXiv:2107.00009 [INSPIRE].
M. Hutchings, F. Morgan, M. Ritoré and A. Ros, Proof of the double bubble conjecture, Annals Math. 155 (2002) 459.
A. Cotton and D. Freeman, The double bubble problem in spherical and hyperbolic space, Int. J. Math. Math. Sci. 32 (2002) 189517.
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].
H. Maxfield, Entanglement entropy in three dimensional gravity, JHEP 04 (2015) 031 [arXiv:1412.0687] [INSPIRE].
W. Dür, G. Vidal and J.I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62 (2000) 062314 [quant-ph/0005115] [INSPIRE].
C.H. Bennett, S. Popescu, D. Rohrlich, J.A. Smolin and A.V. Thapliyal, Exact and asymptotic measures of multipartite pure-state entanglement, Phys. Rev. A 63 (2000) 012307 [quant-ph/9908073].
M. Walter, D. Gross and J. Eisert, Multi-partite entanglement, arXiv e-prints (2016) [arXiv:1612.02437].
S. de Bone, R. Ouyang, K. Goodenough and D. Elkouss, Protocols for creating and distilling multipartite GHZ states with Bell pairs, arXiv e-prints (2020) [arXiv:2010.12259].
P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
Wikipedia contributors, Stratifolds — Wikipedia, the free encyclopedia, (2021) https://en.wikipedia.org/wiki/Stratifold.
T. Tao, An introduction to measure theory, American Mathematical Society, Providence Rhode Island U.S.A (2011).
Wikipedia contributors, Signed measures — Wikipedia, the free encyclopedia, https://en.wikipedia.org/wiki/Signed_measure (2021).
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: 2107.10276
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. Hyperthreads in holographic spacetimes. J. High Energ. Phys. 2021, 118 (2021). https://doi.org/10.1007/JHEP09(2021)118
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2021)118