Abstract
Generalizing the bit thread formalism, we use convex duality to derive dual flow programs to the bipartite and multipartite holographic entanglement of purification proposals and then prove several inequalities using these constructions. In the multipartite case we find the flows exhibit novel behavior which allows for a constrained flux on the boundary of the homology region. We show this flux can be made distinct from bi-partite terms and reflects the truly multipartite portion of the holographic entanglement of purification.
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References
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, JHEP01 (2018) 098 [arXiv:1709.07424] [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, arXiv:1905.00577 [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].
K. Tamaoka, Entanglement Wedge Cross Section from the Dual Density Matrix, Phys. Rev. Lett. 122 (2019) 141601 [arXiv:1809.09109] [INSPIRE].
M. Miyaji and T. Takayanagi, Surface/State Correspondence as a Generalized Holography, PTEP2015 (2015) 073B03 [arXiv:1503.03542] [INSPIRE].
N. Bao, Minimal Purifications, Wormhole Geometries and the Complexity=Action Proposal, arXiv:1811.03113 [INSPIRE].
W.-Z. Guo, Entanglement of Purification and Projective Measurement in CFT, arXiv:1901.00330 [INSPIRE].
N. Bao, A. Chatwin-Davies, J. Pollack and G.N. Remmen, Towards a Bit Threads Derivation of Holographic Entanglement of Purification, JHEP07 (2019) 152 [arXiv:1905.04317] [INSPIRE].
B.M. Terhal, M. Horodecki, D.W. Leung and D.P. DiVincenzo, The entanglement of purification, J. Math. Phys. 43 (2002) 4286 [quant-ph/0202044].
A. Bhattacharyya, T. Takayanagi and K. Umemoto, Entanglement of Purification in Free Scalar Field Theories, JHEP04 (2018) 132 [arXiv:1802.09545] [INSPIRE].
P. Caputa, M. Miyaji, T. Takayanagi and K. Umemoto, Holographic Entanglement of Purification from Conformal Field Theories, Phys. Rev. Lett.122 (2019) 111601 [arXiv:1812.05268] [INSPIRE].
A. Bhattacharyya, A. Jahn, T. Takayanagi and K. Umemoto, Entanglement of Purification in Many Body Systems and Symmetry Breaking, Phys. Rev. Lett.122 (2019) 201601 [arXiv:1902.02369] [INSPIRE].
S. Bagchi, Monogamy, polygamy, and other properties of entanglement of purification, Phys. Rev.A 91 (2015) 042323.
P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev.D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
M. Headrick, General properties of holographic entanglement entropy, JHEP03 (2014) 085 [arXiv:1312.6717] [INSPIRE].
C.A. Agón, J. De Boer and J.F. Pedraza, Geometric Aspects of Holographic Bit Threads, JHEP05 (2019) 075 [arXiv:1811.08879] [INSPIRE].
M. Ghodrati, X.-M. Kuang, B. Wang, C.-Y. Zhang and Y.-T. Zhou, The connection between holographic entanglement and complexity of purification, arXiv:1902.02475 [INSPIRE].
J. Kudler-Flam, I. MacCormack and S. Ryu, Holographic entanglement contour, bit threads and the entanglement tsunami, J. Phys.A 52 (2019) 325401 [arXiv:1902.04654] [INSPIRE].
K. Umemoto and Y. Zhou, Entanglement of Purification for Multipartite States and its Holographic Dual, JHEP10 (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].
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, (2004), [https://doi.org/10.1017/cbo9780511804441].
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].
D.-H. Du, C.-B. Chen and F.-W. Shu, Bit threads and holographic entanglement of purification, arXiv:1904.06871 [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE].
N. Bao and I.F. Halpern, Holographic Inequalities and Entanglement of Purification, JHEP03 (2018) 006 [arXiv:1710.07643] [INSPIRE].
S.X. Cui, P. Hayden, T. He, M. Headrick, B. Stoica and M. Walter, Bit Threads and Holographic Monogamy, arXiv:1808.05234 [INSPIRE].
N. Bao, A. Chatwin-Davies and G.N. Remmen, Entanglement of Purification and Multiboundary Wormhole Geometries, JHEP02 (2019) 110 [arXiv:1811.01983] [INSPIRE].
A.C. Wall, Maximin Surfaces and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav.31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
M. Headrick and V. Hubeny, Covariant bit threads, to appear.
M. Christandl and A. Winter, “squashed entanglement” — an additive entanglement measure, J. Math. Phys. 45 (2004) 829 [quant-ph/0308088].
R.R. Tucci, Entanglement of distillation and conditional mutual information, quant-ph/0202144.
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Harper, J., Headrick, M. Bit threads and holographic entanglement of purification. J. High Energ. Phys. 2019, 101 (2019). https://doi.org/10.1007/JHEP08(2019)101
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DOI: https://doi.org/10.1007/JHEP08(2019)101