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Stabilizer complexity of Hawking radiation

Ritam Basu*, Onkar Parrikar†, Suprakash Paul‡, and Harshit Rajgadia§

  • *Contact author: ritam.basu@tifr.res.in
  • †Contact author: parrikar@theory.tifr.res.in
  • ‡Contact author: suprakash.paul@tifr.res.in
  • §Contact author: harshit.rajgadia@tifr.res.in
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Phys. Rev. D 114, 046016 – Published 18 August, 2026

DOI: https://doi.org/10.1103/49df-l9n4

Abstract

We study the complexity of Hawking radiation for an evaporating black hole from the perspective of the stabilizer theory of quantum computation. Specifically, we calculate Wigner negativity—a magic monotone that can be interpreted as a measure of the stabilizer complexity, or equivalently, the complexity of classical simulation—in various toy models for evaporating black holes. We first calculate the Wigner negativity of Hawking radiation in the Penington-Shenker-Stanford-Yang (PSSY) model directly using the gravitational path integral, and show that the negativity is O(1) before the Page transition, but becomes exponentially large past the Page transition. We also derive a universal, information theoretic formula for the negativity that interpolates between the two extremes. We then study the Wigner negativity of radiation in a dynamical model of black hole evaporation. In this case, the negativity shows a sharp spike at early times resulting from the coupling between the black hole and the radiation system, but at late times when the system settles down, we find that the negativity satisfies the same universal formula as in the PSSY model. Finally, we also propose a geometric formula for Wigner negativity in general holographic states using intuition from fixed area states and random tensor networks, and argue that a python’s lunch in the entanglement wedge implies a stabilizer complexity that is exponentially large in 18GN times the difference between the areas corresponding to the outermost and minimal extremal surfaces.

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References (94)

  1. S. W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14, 2460 (1976).
  2. D. N. Page, Information in black hole radiation, Phys. Rev. Lett. 71, 3743 (1993).
  3. G. Penington, Entanglement wedge reconstruction and the information paradox, J. High Energy Phys. 09 (2020) 002.
  4. 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.
  5. A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The page curve of Hawking radiation from semiclassical geometry, J. High Energy Phys. 03 (2020) 149.
  6. A. Almheiri, R. Mahajan, and J. Maldacena, Islands outside the horizon, arXiv:1910.11077.
  7. G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole interior, J. High Energy Phys. 03 (2022) 205.
  8. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, J. High Energy Phys. 05 (2020) 013.
  9. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93, 035002 (2021).
  10. M. Headrick, V. E. Hubeny, A. Lawrence, and M. Rangamani, Causality & holographic entanglement entropy, J. High Energy Phys. 12 (2014) 162.
  11. B. Czech, J. L. Karczmarek, F. Nogueira, and M. Van Raamsdonk, The gravity dual of a density matrix, Classical Quantum Gravity 29, 155009 (2012).
  12. A. Almheiri, X. Dong, and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, J. High Energy Phys. 04 (2015) 163.
  13. X. Dong, D. Harlow, and A. C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett. 117, 021601 (2016).
  14. D. Harlow, The Ryu–Takayanagi formula from quantum error correction, Commun. Math. Phys. 354, 865 (2017).
  15. T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, J. High Energy Phys. 07 (2017) 151.
  16. J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle, and M. Walter, Entanglement wedge reconstruction via universal recovery channels, Phys. Rev. X 9, 031011 (2019).
  17. V. Balasubramanian, D. Marolf, and M. Rozali, Information recovery from black holes, Gen. Relativ. Gravit. 38, 1529 (2006).
  18. S. Raju, Is holography implicit in canonical gravity? Int. J. Mod. Phys. D 28, 1944011 (2019).
  19. C. Chowdhury, O. Papadoulaki, and S. Raju, A physical protocol for observers near the boundary to obtain bulk information in quantum gravity, SciPost Phys. 10, 106 (2021).
  20. D. Harlow and P. Hayden, Quantum computation vs. firewalls, J. High Energy Phys. 06 (2013) 085.
  21. A. R. Brown, H. Gharibyan, G. Penington, and L. Susskind, The Python’s lunch: Geometric obstructions to decoding Hawking radiation, J. High Energy Phys. 08 (2020) 121.
  22. I. H. Kim, E. Tang, and J. Preskill, The ghost in the radiation: robust encodings of the black hole interior, J. High Energy Phys. 06 (2020) 031.
  23. C. Akers, N. Engelhardt, D. Harlow, G. Penington, and S. Vardhan, The black hole interior from non-isometric codes and complexity, J. High Energy Phys. 06 (2024) 155.
  24. L. Yang and N. Engelhardt, The complexity of learning (pseudo)random dynamics of black holes and other chaotic systems, J. High Energy Phys. 03 (2025) 153.
  25. V. Balasubramanian, A. Kar, C. Li, and O. Parrikar, Quantum error correction in the black hole interior, J. High Energy Phys. 07 (2022) 189.
  26. V. Balasubramanian, A. Kar, C. Li, O. Parrikar, and H. Rajgadia, Quantum error correction from complexity in Brownian SYK, J. High Energy Phys. 08 (2023) 071.
  27. N. Engelhardt, G. Penington, and A. Shahbazi-Moghaddam, A world without pythons would be so simple, Classical Quantum Gravity 38, 234001 (2021).
  28. N. Engelhardt, G. Penington, and A. Shahbazi-Moghaddam, Finding pythons in unexpected places, Classical Quantum Gravity 39, 094002 (2022).
  29. A. May, S. Pasterski, C. Waddell, and M. Xu, Cryptographic tests of the Python’s lunch conjecture, SciPost Phys. 19, 084 (2025).
  30. G. Arora, M. Headrick, A. Lawrence, M. Sasieta, and C. Wolfe, Geometric surprises in the Python’s lunch conjecture, SciPost Phys. 16, 152 (2024).
  31. N. Engelhardt, G. Penington, and A. Shahbazi-Moghaddam, Twice upon a time: Timelike-separated quantum extremal surfaces, J. High Energy Phys. 01 (2024) 033.
  32. D. Gottesman, The Heisenberg representation of quantum computers, arXiv:quant-ph/9807006.
  33. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  34. A. Mari and J. Eisert, Positive Wigner functions render classical simulation of quantum computation efficient, Phys. Rev. Lett. 109, 230503 (2012).
  35. S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).
  36. V. Veitch, C. Ferrie, D. Gross, and J. Emerson, Negative quasi-probability as a resource for quantum computation, New J. Phys. 14, 113011 (2012).
  37. V. Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, The resource theory of stabilizer quantum computation, New J. Phys. 16, 013009 (2014).
  38. W. K. Wootters, A Wigner-function formulation of finite-state quantum mechanics, Ann. Phys. (N.Y.) 176, 1 (1987).
  39. U. Leonhardt, Quantum-state tomography and discrete Wigner function, Phys. Rev. Lett. 74, 4101 (1995).
  40. S. Heiss and S. Weigert, Discrete Moyal-type representations for a spin, Phys. Rev. A 63, 012105 (2000).
  41. C. Miquel, J. P. Paz, and M. Saraceno, Quantum computers in phase space, Phys. Rev. A 65, 062309 (2002).
  42. K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Discrete phase space based on finite fields, Phys. Rev. A 70, 062101 (2004).
  43. D. Gross, Hudson’s theorem for finite-dimensional quantum systems, J. Math. Phys. (N.Y.) 47, 122107 (2006).
  44. C. Cormick, E. F. Galvão, D. Gottesman, J. P. Paz, and A. O. Pittenger, Classicality in discrete Wigner functions, Phys. Rev. A 73, 012301 (2006).
  45. H. Pashayan, J. J. Wallman, and S. D. Bartlett, Estimating outcome probabilities of quantum circuits using quasiprobabilities, Phys. Rev. Lett. 115, 070501 (2015).
  46. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932).
  47. X. Wang, M. M. Wilde, and Y. Su, Quantifying the magic of quantum channels, New J. Phys. 21, 103002 (2019).
  48. R. Basu, P. Chowdhury, A. Ganguly, S. Nath, O. Parrikar, and S. Paul, Wigner negativity, random matrices and gravity, J. High Energy Phys. 01 (2026) 106.
  49. E. Hostens, J. Dehaene, and B. D. Moor, Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic, Phys. Rev. A 71, 042315 (2005).
  50. C. D. White, C. Cao, and B. Swingle, Conformal field theories are magical, Phys. Rev. B 103, 075145 (2021).
  51. Z.-W. Liu and A. Winter, Many-body quantum magic, PRX Quantum 3, 020333 (2022).
  52. X. Turkeshi, A. Dymarsky, and P. Sierant, Pauli spectrum and nonstabilizerness of typical quantum many-body states, Phys. Rev. B 111, 054301 (2025).
  53. E. Tirrito, X. Turkeshi, and P. Sierant, Anticoncentration and nonstabilizerness spreading under ergodic quantum dynamics, Phys. Rev. Lett. 135, 220401 (2025).
  54. J. R. Fliss, Knots, links, and long-range magic, J. High Energy Phys. 04 (2021) 090.
  55. C. D. White and J. H. Wilson, Mana in Haar-random states, arXiv:2011.13937.
  56. K. Goto, T. Nosaka, and M. Nozaki, Probing chaos by magic monotones, Phys. Rev. D 106, 126009 (2022).
  57. L. Leone, S. F. E. Oliviero, Y. Zhou, and A. Hamma, Quantum chaos is quantum, Quantum 5, 453 (2021).
  58. L. Chen, R. J. Garcia, K. Bu, and A. Jaffe, Magic of random matrix product states, Phys. Rev. B 109, 174207 (2024).
  59. P. Niroula, C. D. White, Q. Wang, S. Johri, D. Zhu, C. Monroe, C. Noel, and M. J. Gullans, Phase transition in magic with random quantum circuits, Nat. Phys. 20, 1786 (2024).
  60. X. Turkeshi and P. Sierant, Error-resilience phase transitions in encoding-decoding quantum circuits, Phys. Rev. Lett. 132, 140401 (2024).
  61. Y. Zhang and Y. Gu, Quantum magic dynamics in random circuits, npj Quantum Inf. 12, 87 (2026).
  62. S. F. E. Oliviero, L. Leone, S. Lloyd, and A. Hamma, Unscrambling quantum information with clifford decoders, Phys. Rev. Lett. 132, 080402 (2024).
  63. C. Cao, Non-trivial area operators require non-local magic, J. High Energy Phys. 11 (2024) 105.
  64. C. Cao, G. Cheng, A. Hamma, L. Leone, W. Munizzi, and S. F. E. Oliviero, Gravitational back-reaction is magical, PRX Quantum 6, 040375 (2025).
  65. R. Basu, A. Ganguly, S. Nath, and O. Parrikar, Complexity growth and the Krylov-Wigner function, J. High Energy Phys. 05 (2024) 264.
  66. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  67. M. A. Nielsen, A geometric approach to quantum circuit lower bounds, Quantum Inf. Comput. 6, 213 (2006).
  68. M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum computation as geometry, Science 311, 1133 (2006).
  69. L. Susskind, Entanglement is not enough, Fortschr. Phys. 64, 49 (2016).
  70. L. Susskind, Computational complexity and black hole horizons, Fortschr. Phys. 64, 24 (2016).
  71. R. A. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, J. High Energy Phys. 10 (2017) 107.
  72. S. Chapman, M. P. Heller, H. Marrochio, and F. Pastawski, Toward a definition of complexity for quantum field theory states, Phys. Rev. Lett. 120, 121602 (2018).
  73. A. R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97, 086015 (2018).
  74. V. Balasubramanian, M. Decross, A. Kar, and O. Parrikar, Quantum complexity of time evolution with chaotic hamiltonians, J. High Energy Phys. 01 (2020) 134.
  75. F. G. S. L. Brandão, W. Chemissany, N. Hunter-Jones, R. Kueng, and J. Preskill, Models of quantum complexity growth, PRX Quantum 2, 030316 (2021).
  76. J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eisert, and N. Y. Halpern, Linear growth of quantum circuit complexity, Nat. Phys. 18, 528 (2022).
  77. V. Balasubramanian, M. DeCross, A. Kar, Y. C. Li, and O. Parrikar, Complexity growth in integrable and chaotic models, J. High Energy Phys. 07 (2021) 011.
  78. A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  79. V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
  80. P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in Krylov space: Methods and applications, Phys. Rep. 1125–1128, 1 (2025).
  81. S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M. P. Heller, and N. Y. Halpern, Quantum complexity in gravity, quantum field theory, and quantum information science, Phys. Rep. 1159, 1 (2026).
  82. P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115.
  83. A. Hamilton, D. N. Kabat, G. Lifschytz, and D. A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74, 066009 (2006).
  84. N. Engelhardt and A. C. Wall, Coarse graining holographic black holes, J. High Energy Phys. 05 (2019) 160.
  85. J. Chandra and T. Hartman, Coarse graining pure states in AdS/CFT, J. High Energy Phys. 10 (2023) 030.
  86. P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter, and Z. Yang, Holographic duality from random tensor networks, J. High Energy Phys. 11 (2016) 009.
  87. J. Chandra and T. Hartman, Toward random tensor networks and holographic codes in CFT, J. High Energy Phys. 05 (2023) 109.
  88. H. Geng, L.-Y. Hung, and Y. Jiang, It from ETH: Multi-interval entanglement and replica wormholes from Large-c BCFT ensemble, arXiv:2505.20385.
  89. N. Bao, H. Geng, and Y. Jiang, Ryu-Takayanagi formula for multi-boundary black holes from 2D large-c CFT ensemble, J. High Energy Phys. 10 (2025) 042.
  90. X. Dong, D. Harlow, and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, J. High Energy Phys. 10 (2019) 240.
  91. C. Akers and P. Rath, Holographic Renyi entropy from quantum error correction, J. High Energy Phys. 05 (2019) 052.
  92. P. Caputa, J. Kruthoff, and O. Parrikar, Building tensor networks for holographic states, J. High Energy Phys. 05 (2021) 009; 09 (2022) 112(E).
  93. O. Parrikar, H. Rajgadia, V. Singh, and J. Sorce, Relational bulk reconstruction from modular flow, J. High Energy Phys. 07 (2024) 138.
  94. R. Basu, On the stabilizer complexity of Hawking radiation, GitHub repository (2026), https://github.com/ritambasu61/Onthe- stabilizer-complexity-of-Hawking-radiation.

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Phys. Rev. D 114, 046016– Published 18 August, 2026
  • Received 4 March 2026
  • Accepted 13 July 2026
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