- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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

Coherence-Induced Deep Thermalization Transition in Random Permutation Quantum Dynamics

Chang Liu1, Matteo Ippoliti2, and Wen Wei Ho1,3,*

  • *Contact author: wenweiho@nus.edu.sg
PDF

Phys. Rev. Lett. 136, 100404 – Published 12 March, 2026

DOI: https://doi.org/10.1103/mjgs-y2z9

Abstract

We report a phase transition in the projected ensemble—the collection of postmeasurement wave functions of a local subsystem obtained by measuring its complement. The transition emerges in systems undergoing random permutation dynamics, a type of quantum time evolution wherein computational basis states are shuffled without creating superpositions. It separates a phase exhibiting deep thermalization, where the projected ensemble is distributed over Hilbert space in a maximally entropic fashion (Haar random), from a phase where it is minimally entropic (“classical bit-string ensemble”). Crucially, this deep thermalization transition is invisible to the subsystem’s density matrix, which always exhibits thermalization to infinite temperature across the phase diagram. Through a combination of analytical arguments and numerical simulations, we show that the transition is tuned by the total amount of coherence injected by the input state and the measurement basis, and is exhibited robustly across different microscopic models. Our findings represent a novel form of ergodicity-breaking universality in quantum many-body dynamics, characterized not by a failure of regular thermalization, but rather by a failure of deep thermalization.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (94)

  1. M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008).
  2. J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015).
  3. R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  4. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019).
  5. L. M. Sieberer, M. Buchhold, J. Marino, and S. Diehl, Rev. Mod. Phys. 97, 025004 (2025).
  6. Y. Sekino and L. Susskind, J. High Energy Phys. 2008 (10), 065.
  7. S. H. Shenker and D. Stanford, J. High Energy Phys. 2014 (03), 067.
  8. J. Maldacena, S. H. Shenker, and D. Stanford, J. High Energy Phys. 2016 (08), 106.
  9. P. Hayden and J. Preskill, J. High Energy Phys. 2007 (09), 120.
  10. M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Annu. Rev. Condens. Matter Phys. 14, 335 (2023).
  11. J. Choi, A. L. Shaw, I. S. Madjarov, X. Xie, R. Finkelstein, J. P. Covey, J. S. Cotler, D. K. Mark, H.-Y. Huang, A. Kale, H. Pichler, F. Brandao, S. Choi, and M. Endres, Nature (London) 613, 468 (2023).
  12. J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernández, J. Choi, A. L. Shaw, M. Endres, and S. Choi, PRX Quantum 4, 010311 (2023).
  13. W. W. Ho and S. Choi, Phys. Rev. Lett. 128, 060601 (2022).
  14. D. K. Mark, F. Surace, A. Elben, A. L. Shaw, J. Choi, G. Refael, M. Endres, and S. Choi, Phys. Rev. X 14, 041051 (2024).
  15. C. Liu, Q. C. Huang, and W. W. Ho, Phys. Rev. Lett. 133, 260401 (2024).
  16. R.-A. Chang, H. Shrotriya, W. W. Ho, and M. Ippoliti, PRX Quantum 6, 020343 (2025).
  17. M. Ippoliti and W. W. Ho, Quantum 6, 886 (2022).
  18. M. Ippoliti and W. W. Ho, PRX Quantum 4, 030322 (2023).
  19. M. McGinley and T. Schuster, arXiv:2511.17172.
  20. W.-K. Mok, T. Haug, W. W. Ho, and J. Preskill, arXiv:2601.00266.
  21. M. Lucas, L. Piroli, J. De Nardis, and A. De Luca, Phys. Rev. A 107, 032215 (2023).
  22. M. Bejan, B. Béri, and M. McGinley, Phys. Rev. Lett. 135, 020401 (2025).
  23. P. W. Claeys and A. Lamacraft, Quantum 6, 738 (2022).
  24. T. Bhore, J.-Y. Desaules, and Z. Papić, Phys. Rev. B 108, 104317 (2023).
  25. A. Chan and A. De Luca, J. Phys. A 57, 405001 (2024).
  26. N. D. Varikuti and S. Bandyopadhyay, Quantum 8, 1456 (2024).
  27. H. Shrotriya and W. W. Ho, SciPost Phys. 18, 107 (2025).
  28. B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Nat. Commun. 16, 6341 (2025).
  29. X.-H. Yu, W. W. Ho, and P. Kos, Phys. Rev. Lett. 135, 260402 (2025).
  30. A. Sherry and S. Roy, arXiv:2507.14135.
  31. S. Chakraborty, S. Choi, S. Ghosh, and T. Giurgică-Tiron, Phys. Rev. Lett. 135, 210603 (2025).
  32. M. Aldana, S. Coppersmith, and L. P. Kadanoff, Boolean dynamics with random couplings, in Perspectives and Problems in Nonlinear Science (Springer, New York, 2003), pp. 23–89.
  33. B. Bertini, K. Klobas, P. Kos, and D. Malz, Phys. Rev. X 15, 011015 (2025).
  34. D. Szász-Schagrin, M. Mazzoni, B. Bertini, K. Klobas, and L. Piroli, arXiv:2505.06158.
  35. B. Bertini, K. Klobas, P. Kos, and D. Malz, arXiv:2508.10890.
  36. T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
  37. A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016).
  38. A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys. 89, 041003 (2017).
  39. A. Hickey and G. Gour, J. Phys. A 51, 414009 (2018).
  40. K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Phys. Rev. A 103, 032401 (2021).
  41. S. Bravyi and J. Haah, Phys. Rev. A 86, 052329 (2012).
  42. M. Howard and E. Campbell, Phys. Rev. Lett. 118, 090501 (2017).
  43. Z.-W. Liu and A. Winter, PRX Quantum 3, 020333 (2022).
  44. Q. Zhuang, P. W. Shor, and J. H. Shapiro, Phys. Rev. A 97, 052317 (2018).
  45. L. Lami, B. Regula, X. Wang, R. Nichols, A. Winter, and G. Adesso, Phys. Rev. A 98, 022335 (2018).
  46. W. T. Gowers, Comb. Probab. Comput. 5, 119 (1996).
  47. S. Hoory, A. Magen, S. Myers, and C. Rackoff, Theor. Comput. Sci. 348, 251 (2005).
  48. A. Brodsky and S. Hoory, Random Struct. Algorithms 32, 274 (2008).
  49. S. Gopalakrishnan and B. Zakirov, Quantum Sci. Technol. 3, 044004 (2018).
  50. J. Iaconis, S. Vijay, and R. Nandkishore, Phys. Rev. B 100, 214301 (2019).
  51. J. Iaconis, A. Lucas, and X. Chen, Phys. Rev. B 102, 224311 (2020).
  52. J. Iaconis, PRX Quantum 2, 010329 (2021).
  53. X. Feng and M. Ippoliti, J. High Energy Phys. 2025 (02), 128.
  54. W. Lee, H. Kwon, and G. Y. Cho, arXiv:2411.03974.
  55. See Supplemental Material at http://link.aps.org/supplemental/10.1103/mjgs-y2z9 for details on statements, proofs of theorems, and numerics presented in the main text, which includes Ref. [56].
  56. S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975).
  57. B. Collins and M. Nagatsu, arXiv:2503.18453.

  58. This is because the class of generator states (RPSs in this case) is invariant under qubit swaps in B.

  59. R. Jozsa, D. Robb, and W. K. Wootters, Phys. Rev. A 49, 668 (1994).
  60. A. Touil, F. Anza, S. Deffner, and J. P. Crutchfield, Quantum 8, 1494 (2024).
  61. U. Singh, L. Zhang, and A. K. Pati, Phys. Rev. A 93, 032125 (2016).

  62. Asymptotically, this grows as lndA+γ−1 with corrections in 1/dA, where γ is the Euler-Mascheroni constant.

  63. We emphasize that this is not the same as the coherence of the average state Cr([EψA∼EPE|ψA⟩⟨ψA|]), since coherence is not a linear function of the state.

  64. Generalized to include imaginary phases.

  65. S. Aaronson, A. Bouland, B. Fefferman, S. Ghosh, U. Vazirani, C. Zhang, and Z. Zhou, in 15th Innovations in Theoretical Computer Science Conference (ITCS 2024), Leibniz International Proceedings in Informatics (LIPIcs). Vol. 287, edited by V. Guruswami (Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2024), pp. 2:1–2:21.
  66. T. Giurgica-Tiron and A. Bouland, arXiv:2312.09206.
  67. F. G. Jeronimo, N. Magrafta, and P. Wu, arXiv:2312.15285.
  68. D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys. (Amsterdam) 321, 1126 (2006).
  69. I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. Lett. 95, 206603 (2005).
  70. R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  71. F. Alet and N. Laflorencie, C. R. Phys. 19, 498 (2018).
  72. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019).
  73. V. Oganesyan and D. A. Huse, Phys. Rev. B 75, 155111 (2007).
  74. M. Žnidarič, T. Prosen, and P. Prelovšek, Phys. Rev. B 77, 064426 (2008).
  75. H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner et al., Nature (London) 551, 579 (2017).
  76. S. Moudgalya, N. Regnault, and B. A. Bernevig, Phys. Rev. B 98, 235156 (2018).
  77. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Nat. Phys. 14, 745 (2018).
  78. S. Choi, C. J. Turner, H. Pichler, W. W. Ho, A. A. Michailidis, Z. Papić, M. Serbyn, M. D. Lukin, and D. A. Abanin, Phys. Rev. Lett. 122, 220603 (2019).
  79. W. W. Ho, S. Choi, H. Pichler, and M. D. Lukin, Phys. Rev. Lett. 122, 040603 (2019).
  80. C.-J. Lin and O. I. Motrunich, Phys. Rev. Lett. 122, 173401 (2019).
  81. S. Scherg, T. Kohlert, P. Sala, F. Pollmann, B. Hebbe Madhusudhana, I. Bloch, and M. Aidelsburger, Nat. Commun. 12, 4490 (2021).
  82. G.-X. Su, H. Sun, A. Hudomal, J.-Y. Desaules, Z.-Y. Zhou, B. Yang, J. C. Halimeh, Z.-S. Yuan, Z. Papić, and J.-W. Pan, Phys. Rev. Res. 5, 023010 (2023).
  83. D. Bhowmick, V. B. Bulchandani, and W. W. Ho, arXiv:2505.05435.
  84. Y. Li, X. Chen, and Matthew P. A. Fisher, Phys. Rev. B 98, 205136 (2018).
  85. Y. Li, X. Chen, and Matthew P. A. Fisher, Phys. Rev. B 100, 134306 (2019).
  86. B. Skinner, J. Ruhman, and A. Nahum, Phys. Rev. X 9, 031009 (2019).
  87. A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Phys. Rev. B 99, 224307 (2019).
  88. Y. Bao, S. Choi, and E. Altman, Phys. Rev. B 101, 104301 (2020).
  89. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010).
  90. A. S. Holevo, Prob. Peredachi Inf. 9, 3 (1973), https://www.mathnet.ru/eng/ppi903.
  91. H. Lóio, G. Lami, L. Leone, M. McGinley, X. Turkeshi, and J. D. Nardis, arXiv:2510.13950.
  92. V. Shende, A. Prasad, I. Markov, and J. Hayes, IEEE Transact. Comput.-Aided Des. Integr. Circuits Syst. 22, 710 (2003).
  93. C.-F. Chen, J. Haah, J. Haferkamp, Y. Liu, T. Metger, and X. Tan, arXiv:2406.07478.
  94. W. Gay, W. He, N. Kocurek, and R. O’Donnell, in Annual International Cryptology Conference (Springer, New York, 2025), pp. 651–678.

Outline

Information

Phys. Rev. Lett. 136, 100404– Published 12 March, 2026
  • Received 15 November 2025
  • Accepted 18 February 2026
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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation