Abstract
The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a “bit thread” formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for S(AB) in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.
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References
B. Swingle, Entanglement renormalization and holography, Phys. Rev.D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav.42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
M. Miyaji and T. Takayanagi, Surface/state correspondence as a generalized holography, PTEP2015 (2015) 073B03 [arXiv:1503.03542].
M. Miyaji et al., Continuous multiscale entanglement renormalization ansatz as holographic surface-state correspondence, Phys. Rev. Lett.115 (2015) 171602.
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
P. Hayden et al., Holographic duality from random tensor networks, JHEP11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
M. Freedman and M. Headrick, Bit threads and holographic entanglement, Commun. Math. Phys.352 (2017) 407 [arXiv:1604.00354] [INSPIRE].
M. Miyaji, T. Takayanagi and K. Watanabe, From path integrals to tensor networks for the AdS/CFT correspondence, Phys. Rev.D 95 (2017) 066004 [arXiv:1609.04645] [INSPIRE].
P. Caputa et al., Anti-de Sitter space from optimization of path integrals in conformal field theories, Phys. Rev. Lett.119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].
P. Caputa et al., Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT, JHEP11 (2017) 097 [arXiv:1706.07056] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [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.E. Hubeny, Riemannian and Lorentzian flow-cut theorems, Class. Quant. Grav.35 (2018) 10 [arXiv:1710.09516] [INSPIRE].
J. Harper, M. Headrick and A. Rolph, Bit threads in higher curvature gravity, JHEP11 (2018)168 [arXiv:1807.04294] [INSPIRE].
S.X. Cui et al., Bit threads and holographic monogamy, Commun. Math. Phys. (2019) [arXiv:1808.05234] [INSPIRE].
V.E. Hubeny, Bulk locality and cooperative flows, JHEP12 (2018) 068 [arXiv:1808.05313] [INSPIRE].
C.A. Agón, J. De Boer and J.F. Pedraza, Geometric aspects of holographic bit threads, JHEP05 (2019) 075 [arXiv:1811.08879] [INSPIRE].
I. Bakhmatov et al., Calibrated entanglement entropy, JHEP07 (2017) 117 [arXiv:1705.08319] [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].
T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys.14 (2018) 573 [arXiv:1708.09393] [INSPIRE].
P. Nguyen et al., Entanglement of purification: from spin chains to holography, JHEP01 (2018) 098 [arXiv:1709.07424] [INSPIRE].
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav.29 (2012) 155009 [arXiv:1204.1330] [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].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
N. Bao and I.F. Halpern, Holographic inequalities and entanglement of purification, JHEP03 (2018) 006 [arXiv:1710.07643] [INSPIRE].
A. Bhattacharyya, T. Takayanagi and K. Umemoto, Entanglement of purification in free scalar field theories, JHEP04 (2018) 132 [arXiv:1802.09545] [INSPIRE].
D. Blanco, M. Leston and G. Pérez-Nadal, Gravity from entanglement for boundary subregions, JHEP18 (2018) 130 [arXiv:1803.01874] [INSPIRE].
H. Hirai, K. Tamaoka and T. Yokoya, Towards entanglement of purification for conformal field theories, PTEP2018 (2018) 063B03 [arXiv:1803.10539] [INSPIRE].
R. Espíndola, A. Guijosa and J.F. Pedraza, Entanglement wedge reconstruction and entanglement of purification, Eur. Phys. J.C 78 (2018) 646 [arXiv:1804.05855] [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].
Y. Nomura, P. Rath and N. Salzetta, Pulling the boundary into the bulk, Phys. Rev.D 98 (2018) 026010 [arXiv:1805.00523] [INSPIRE].
K. Umemoto and Y. Zhou, Entanglement of purification for multipartite states and its holographic dual, JHEP10 (2018) 152 [arXiv:1805.02625] [INSPIRE].
R. Abt et al., Holographic subregion complexity from kinematic space, JHEP01 (2019) 012 [arXiv:1805.10298] [INSPIRE].
A. May and E. Hijano, The holographic entropy zoo, JHEP10 (2018) 036 [arXiv:1806.06077] [INSPIRE].
Y. Chen, X. Dong, A. Lewkowycz and X.-L. Qi, Modular flow as a disentangler, JHEP12 (2018) 083 [arXiv:1806.09622] [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].
J.C. Cresswell, I.T. Jardine and A.W. Peet, Holographic relations for OPE blocks in excited states, JHEP03 (2019) 058 [arXiv:1809.09107] [INSPIRE].
E. Caceres and M.-L. Xiao, Complexity-action of subregions with corners, JHEP03 (2019) 062 [arXiv:1809.09356] [INSPIRE].
R.-Q. Yang, C.-Y. Zhang and W.-M. Li, Holographic entanglement of purification for thermofield double states and thermal quench, JHEP01 (2019) 114 [arXiv:1810.00420] [INSPIRE].
N. Bao, A. Chatwin-Davies and G.N. Remmen, Entanglement of purification and multiboundary wormhole geometries, JHEP02 (2019) 110 [arXiv:1811.01983] [INSPIRE].
N. Bao, Minimal purifications, wormhole geometries and the complexity=action proposal, arXiv:1811.03113 [INSPIRE].
N. Bao, G. Penington, J. Sorce and A.C. Wall, Beyond toy models: distilling tensor networks in full AdS/CFT, arXiv:1812.01171 [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].
W.-Z. Guo, Entanglement of purification and projective measurement in CFT, arXiv:1901.00330 [INSPIRE].
P. Liu, Y. Ling, C. Niu and J.-P. Wu, Entanglement of purification in holographic systems, arXiv:1902.02243 [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].
M. Ghodrati et al., 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. Babaei Velni, M.R. Mohammadi Mozaffar and M.H. Vahidinia, Some aspects of entanglement wedge cross-section, JHEP05 (2019) 200 [arXiv:1903.08490] [INSPIRE].
M. Horodecki and M. Piani, On quantum advantage in dense coding, J. Phys.A 45 (2012) 105306 [quant-ph/0701134].
T. Hirata and T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy, JHEP02 (2007) 042 [hep-th/0608213] [INSPIRE].
T. Nishioka and T. Takayanagi, AdS bubbles, entropy and closed string tachyons, JHEP01 (2007) 090 [hep-th/0611035] [INSPIRE].
I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys.B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev.D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
A. Frank, A.V. Karzanov and A. Sebo, On integer multiflow maximization, SIAM J. Discr. Math.10 (1997) 158.
A. Schrijver, Combinatorial optimization: polyhedra and efficiency, Springer, Germany (2003).
S. Bagchi and A.K. Pati, Monogamy, polygamy, and other properties of entanglement of purification, Phys. Rev.A 91 (2015) 042323 [arXiv:1502.01272].
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Du, DH., Chen, CB. & Shu, FW. Bit threads and holographic entanglement of purification. J. High Energ. Phys. 2019, 140 (2019). https://doi.org/10.1007/JHEP08(2019)140
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DOI: https://doi.org/10.1007/JHEP08(2019)140