Abstract
In holographic theories, the reflected entropy has been shown to be dual to the area of the entanglement wedge cross section. We study the same problem in random tensor networks demonstrating an equivalent duality. For a single random tensor we analyze the important non-perturbative effects that smooth out the discontinuity in the reflected entropy across the Page phase transition. By summing over all such effects, we obtain the reflected entanglement spectrum analytically, which agrees well with numerical studies. This motivates a prescription for the analytic continuation required in computing the reflected entropy and its Rényi generalization which resolves an order of limits issue previously identified in the literature. We apply this prescription to hyperbolic tensor networks and find answers consistent with holographic expectations. In particular, the random tensor network has the same non-trivial tripartite entanglement structure expected from holographic states. We furthermore show that the reflected Rényi spectrum is not flat, in sharp contrast to the usual Rényi spectrum of these networks. We argue that the various distinct contributions to the reflected entanglement spectrum can be organized into approximate superselection sectors. We interpret this as resulting from an effective description of the canonically purified state as a superposition of distinct tensor network states. Each network is constructed by doubling and gluing various candidate entanglement wedges of the original network. The superselection sectors are labelled by the different cross-sectional areas of these candidate entanglement wedges.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
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].
N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J.M. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
C. Akers and P. Rath, Holographic Renyi Entropy from Quantum Error Correction, JHEP 05 (2019) 052 [arXiv:1811.05171] [INSPIRE].
X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, JHEP 10 (2019) 240 [arXiv:1811.05382] [INSPIRE].
X. Dong and D. Marolf, One-loop universality of holographic codes, JHEP 03 (2020) 191 [arXiv:1910.06329] [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178 [arXiv:1905.00577] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
P. Hayden, O. Parrikar and J. Sorce, The Markov gap for geometric reflected entropy, JHEP 10 (2021) 047 [arXiv:2107.00009] [INSPIRE].
T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys. 14 (2018) 573 [arXiv:1708.09393] [INSPIRE].
A. Lewkowycz and J.M. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [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].
C. Akers, N. Engelhardt, G. Penington and M. Usatyuk, Quantum Maximin Surfaces, JHEP 08 (2020) 140 [arXiv:1912.02799] [INSPIRE].
C. Akers and P. Rath, Entanglement Wedge Cross Sections Require Tripartite Entanglement, JHEP 04 (2020) 208 [arXiv:1911.07852] [INSPIRE].
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].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
C. Akers and G. Penington, Leading order corrections to the quantum extremal surface prescription, JHEP 04 (2021) 062 [arXiv:2008.03319] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
H. Shapourian, S. Liu, J. Kudler-Flam and A. Vishwanath, Entanglement Negativity Spectrum of Random Mixed States: A Diagrammatic Approach, PRX Quantum 2 (2021) 030347 [arXiv:2011.01277] [INSPIRE].
D. Marolf, CFT sewing as the dual of AdS cut-and-paste, JHEP 02 (2020) 152 [arXiv:1909.09330] [INSPIRE].
Y. Kusuki and K. Tamaoka, Entanglement Wedge Cross Section from CFT: Dynamics of Local Operator Quench, JHEP 02 (2020) 017 [arXiv:1909.06790] [INSPIRE].
C. Akers, T. Faulkner, S. Lin and P. Rath, The Page Curve for Reflected Entropy, arXiv:2201.11730 [INSPIRE].
A.W. Harrow, The church of the symmetric subspace, arXiv:1308.6595.
N. Engelhardt and A.C. Wall, Decoding the Apparent Horizon: Coarse-Grained Holographic Entropy, Phys. Rev. Lett. 121 (2018) 211301 [arXiv:1706.02038] [INSPIRE].
N. Engelhardt and A.C. Wall, Coarse Graining Holographic Black Holes, JHEP 05 (2019) 160 [arXiv:1806.01281] [INSPIRE].
X. Dong, S. McBride and W.W. Weng, Replica Wormholes and Holographic Entanglement Negativity, arXiv:2110.11947 [INSPIRE].
S. Vardhan, J. Kudler-Flam, H. Shapourian and H. Liu, Bound entanglement in thermalized states and black hole radiation, arXiv:2110.02959 [INSPIRE].
S. Fischetti and D. Marolf, Complex Entangling Surfaces for AdS and Lifshitz Black Holes?, Class. Quant. Grav. 31 (2014) 214005 [arXiv:1407.2900] [INSPIRE].
X. Dong and H. Wang, Enhanced corrections near holographic entanglement transitions: a chaotic case study, JHEP 11 (2020) 007 [arXiv:2006.10051] [INSPIRE].
D. Marolf, S. Wang and Z. Wang, Probing phase transitions of holographic entanglement entropy with fixed area states, JHEP 12 (2020) 084 [arXiv:2006.10089] [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].
S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
M. Fannes, A continuity property of the entropy density for spin lattice systems, Commun. Math. Phys. 31 (1973) 291.
K.M. Audenaert, A sharp fannes-type inequality for the von Neumann entropy, quant-ph/0610146.
A.E. Rastegin, Some general properties of unified entropies, J. Stat. Phys. 143 (2011) 1120.
E. Lubkin, Entropy of an n-system from its correlation with a k-reservoir, J. Math. Phys. 19 (1978) 1028.
Z.-W. Liu, S. Lloyd, E.Y. Zhu and H. Zhu, Entanglement, quantum randomness, and complexity beyond scrambling, JHEP 07 (2018) 041 [arXiv:1703.08104] [INSPIRE].
J. Jurkiewicz, G. Lukaszewski and M. Nowak, Diagrammatic approach to fluctuations in the wishart ensemble, Acta Phys. Pol. B 39 (2008) 799.
E. Brézin and A. Zee, Universality of the correlations between eigenvalues of large random matrices, Nucl. Phys. B 402 (1993) 613 [INSPIRE].
E. Brézin and A. Zee, Universal relation between Green’s functions in random matrix theory, Nucl. Phys. B 453 (1995) 531 [cond-mat/9507032] [INSPIRE].
J. Kudler-Flam, V. Narovlansky and S. Ryu, Negativity spectra in random tensor networks and holography, JHEP 02 (2022) 076 [arXiv:2109.02649] [INSPIRE].
C. Akers, T. Faulkner, S. Lin and P. Rath, to appear.
D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [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].
X.-L. Qi, Z. Yang and Y.-Z. You, Holographic coherent states from random tensor networks, JHEP 08 (2017) 060 [arXiv:1703.06533] [INSPIRE].
A. Almheiri, X. Dong and B. Swingle, Linearity of Holographic Entanglement Entropy, JHEP 02 (2017) 074 [arXiv:1606.04537] [INSPIRE].
T. Faulkner, The holographic map as a conditional expectation, arXiv:2008.04810 [INSPIRE].
E. Shaghoulian, Emergent gravity from Eguchi-Kawai reduction, JHEP 03 (2017) 011 [arXiv:1611.04189] [INSPIRE].
W. Donnelly, B. Michel, D. Marolf and J. Wien, Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers, JHEP 04 (2017) 093 [arXiv:1611.05841] [INSPIRE].
P. Biane, Some properties of crossings and partitions, Discret. Math. 175 (1997) 41.
G. Birkhoff, Lattice Theory, third edition, American Mathematical Society, Providence, RI, U.S.A. (1967).
J.S. Kim, S. Seo and H. Shin, Annular noncrossing permutations and minimal transitive factorizations, J. Combin. Theor. A 124 (2012) 251.
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: 2112.09122
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
Akers, C., Faulkner, T., Lin, S. et al. Reflected entropy in random tensor networks. J. High Energ. Phys. 2022, 162 (2022). https://doi.org/10.1007/JHEP05(2022)162
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2022)162