- 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

Mixed State Deep Thermalization

Xie-Hang Yu1, Wen Wei Ho2,3, and Pavel Kos1

PDF

Phys. Rev. Lett. 135, 260402 – Published 29 December, 2025

DOI: https://doi.org/10.1103/t6zs-3f8k

Abstract

We introduce the notion of the mixed state projected ensemble (MSPE), a collection of mixed states describing a local region of a quantum many-body system, conditioned upon measurements of the complementary region which are incomplete. This constitutes a generalization of the pure state projected ensemble in which measurements are assumed ideal and complete, and which has been shown to tend towards limiting pure state distributions depending only on symmetries of the system, thus representing a new kind of universality in quantum equilibration dubbed deep thermalization. We study the MSPE generated by solvable (1+1)d dual-unitary quantum circuit evolution, and identify the limiting mixed state distributions which emerge at late times depending on the size of the incomplete measurement, which we assume to be lossy, finding that they correspond to certain random density matrix ensembles known in the literature. We also derive the rate of the emergence of such universality. Furthermore, we investigate the quantum information properties of the states composing the ensemble, specifically their capacity to teleport quantum information between the ends of the system. The teleportation fidelity is upper bounded by the quantum conditional entropy, which we find exhibits a sharp transition from zero to maximal when the number of measurements lost matches of that the number of degrees of freedom to be teleported. Our results initiate the first investigation of deep thermalization for mixed state ensembles, relevant for present-day quantum simulation experiments wherein measurements are typically not perfect, and also amount to a physical and natural way of sampling from hitherto abstract random density matrix ensembles.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (67)

  1. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  2. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  3. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  4. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  5. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  6. 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 et al., Emergent quantum randomness and benchmarking from Hamiltonian many-body dynamics, Nature (London) 613, 468 (2023).
  7. S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature (London) 622, 268 (2023).
  8. C. Gross and W. S. Bakr, Quantum gas microscopy for single atom and spin detection, Nat. Phys. 17, 1316 (2021).
  9. D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch, and J. Zeiher, Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion, Science 376, 716 (2022).
  10. D. Greif, M. F. Parsons, A. Mazurenko, C. S. Chiu, S. Blatt, F. Huber, G. Ji, and M. Greiner, Site-resolved imaging of a fermionic Mott insulator, Science 351, 953 (2016).
  11. R. Acharya, D. A. Abanin, L. Aghababaie-Beni, I. Aleiner, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. Astrakhantsev et al., Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2024).
  12. S. A. Moses et al., A race-track trapped-ion quantum processor, Phys. Rev. X 13, 041052 (2023).
  13. M. Liu, R. Shaydulin, P. Niroula, M. DeCross, S.-H. Hung, W. Y. Kon, E. Cervero-Martín, K. Chakraborty, O. Amer, S. Aaronson et al., Certified randomness using a trapped-ion quantum processor, Nature (London) 640, 343 (2025).
  14. 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 et al., Preparing random states and benchmarking with many-body quantum chaos, Nature (London) 613, 468 (2023).
  15. J. S. Cotler, D. K. Mark, H.-Y. Huang, F. Hernández, J. Choi, A. L. Shaw, M. Endres, and S. Choi, Emergent quantum state designs from individual many-body wave functions, PRX Quantum 4, 010311 (2023).
  16. M. Ippoliti and W. W. Ho, Dynamical purification and the emergence of quantum state designs from the projected ensemble, PRX Quantum 4, 030322 (2023).
  17. D. K. Mark, F. Surace, A. Elben, A. L. Shaw, J. Choi, G. Refael, M. Endres, and S. Choi, Maximum entropy principle in deep thermalization and in Hilbert-space ergodicity, Phys. Rev. X 14, 041051 (2024).
  18. B. Zhang, P. Xu, X. Chen, and Q. Zhuang, Holographic deep thermalization for secure and efficient quantum random state generation, Nat. Commun. 16, 6341 (2025).
  19. W. W. Ho and S. Choi, Exact emergent quantum state designs from quantum chaotic dynamics, Phys. Rev. Lett. 128, 060601 (2022).
  20. M. Ippoliti and W. W. Ho, Solvable model of deep thermalization with distinct design times, Quantum 6, 886 (2022).
  21. R.-A. Chang, H. Shrotriya, W. W. Ho, and M. Ippoliti, Deep thermalization under charge-conserving quantum dynamics, PRX Quantum 6, 020343 (2025).
  22. P. W. Claeys and A. Lamacraft, Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics, Quantum 6, 738 (2022).
  23. H. Shrotriya and W. W. Ho, Nonlocality of deep thermalization, SciPost Phys. 18, 107 (2025).
  24. M. Lucas, L. Piroli, J. De Nardis, and A. De Luca, Generalized deep thermalization for free fermions, Phys. Rev. A 107, 032215 (2023).
  25. C. Liu, Q. C. Huang, and W. W. Ho, Deep thermalization in Gaussian continuous-variable quantum systems, Phys. Rev. Lett. 133, 260401 (2024).
  26. G. Lami, A. D. Luca, X. Turkeshi, and J. D. Nardis, Quantum randomness through emergent confinement mechanism in measured tensor network states, arXiv:2504.16995.
  27. B. Bertini, P. Kos, and T. Prosen, Exact correlation functions for dual-unitary lattice models in 1+1 dimensions, Phys. Rev. Lett. 123, 210601 (2019).
  28. M. Akila, D. Waltner, B. Gutkin, and T. Guhr, Particle-time duality in the kicked Ising spin chain, J. Phys. A 49, 375101 (2016).
  29. S. Gopalakrishnan and A. Lamacraft, Unitary circuits of finite depth and infinite width from quantum channels, Phys. Rev. B 100, 064309 (2019).
  30. L. Piroli, B. Bertini, J. I. Cirac, and T. Prosen, Exact dynamics in dual-unitary quantum circuits, Phys. Rev. B 101, 094304 (2020).
  31. P. W. Claeys and A. Lamacraft, Maximum velocity quantum circuits, Phys. Rev. Res. 2, 033032 (2020).
  32. S. A. Rather, S. Aravinda, and A. Lakshminarayan, Creating ensembles of dual unitary and maximally entangling quantum evolutions, Phys. Rev. Lett. 125, 070501 (2020).
  33. P. Kos, B. Bertini, and T. c. v. Prosen, Correlations in perturbed dual-unitary circuits: Efficient path-integral formula, Phys. Rev. X 11, 011022 (2021).
  34. B. Bertini, P. Kos, and T. Prosen, Random matrix spectral form factor of dual-unitary quantum circuits, Commun. Math. Phys. 387, 597 (2021).
  35. R. Suzuki, K. Mitarai, and K. Fujii, Computational power of one-and two-dimensional dual-unitary quantum circuits, Quantum 6, 631 (2022).
  36. M. L. Mehta, Random Matrices (Elsevier, New York, 1990).
  37. K. Życzkowski, K. A. Penson, I. Nechita, and B. Collins, Generating random density matrices, J. Math. Phys. (N.Y.) 52, 062201 (2011).
  38. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895 (1993).
  39. Y. Bao, M. Block, and E. Altman, Finite-time teleportation phase transition in random quantum circuits, Phys. Rev. Lett. 132, 030401 (2024).

  40. For even t, measuring the rightmost output qudit in the computational basis leaves our results unchanged.

  41. A. Asadian, P. Erker, M. Huber, and C. Klöckl, Heisenberg-Weyl observables: Bloch vectors in phase space, Phys. Rev. A 94, 010301(R) (2016).

  42. This assumption can be relaxed, since Refs. [43, 44] show that a unitary k-design forms exponentially fast in the lengths of B1 and B3, so our results extend to finite B1 and B3, with ε-approximate deep thermalization occurring in chains of length O(log(1/ε)).

  43. R. Suzuki, H. Katsura, Y. Mitsuhashi, T. Soejima, J. Eisert, and N. Yoshioka, More global randomness from less random local gates, arXiv:2410.24127.
  44. J. Riddell, K. Klobas, and B. Bertini, Quantum state designs from minimally random quantum circuits, arXiv:2503.05698.
  45. H.-J. Sommers and K. Życzkowski, Statistical properties of random density matrices, J. Phys. A 37, 8457 (2004).
  46. See Supplemental Material at http://link.aps.org/supplemental/10.1103/t6zs-3f8k for details, which further includes citations [47–49].
  47. T. A. de Lima and M. Ayala-Rincón, Complexity of Cayley distance and other general metrics on permutation groups, in 2012 7th Colombian Computing Congress (CCC) (Institute of Electrical and Electronics Engineers (IEEE), Piscataway, New Jersey (USA), 2012), pp. 1–6.
  48. X.-H. Yu, J. I. Cirac, P. Kos, and G. Styliaris, Dual-isometric projected entangled pair states, Phys. Rev. Lett. 133, 190401 (2024).
  49. J.-L. Brylinski and R. Brylinski, Universal quantum gates, in Mathematics of Quantum Computation (Chapman and Hall/CRC, New York, 2002), pp. 117–134.

  50. For large but finite d, their highly degenerate subleading terms become relevant for moments with k≫d [see Eq. (a7) in the End Matter].

  51. We say more about the distribution of the eigenvalues in Supplemental Material [46].

  52. I. Lovas, U. Agrawal, and S. Vijay, Quantum coding transitions in the presence of boundary dissipation, PRX Quantum 5, 030327 (2024).
  53. N. J. Cerf and C. Adami, Negative entropy and information in quantum mechanics, Phys. Rev. Lett. 79, 5194 (1997).
  54. B. Schumacher and M. A. Nielsen, Quantum data processing and error correction, Phys. Rev. A 54, 2629 (1996).

  55. We refer to the environmental degrees of freedom arising from the purification of tracing out m sites in B2.

  56. C. Neill et al., A blueprint for demonstrating quantum supremacy with superconducting qubits, Science 360, 195 (2018).
  57. A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta, Validating quantum computers using randomized model circuits, Phys. Rev. A 100, 032328 (2019).
  58. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  59. A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, On the complexity and verification of quantum random circuit sampling, Nat. Phys. 15, 159 (2019).
  60. G. Alagic, T. Gagliardoni, and C. Majenz, Unforgeable quantum encryption, in Advances in Cryptology—EUROCRYPT 2018, edited by J. B. Nielsen and V. Rijmen (Springer International Publishing, Cham, 2018), pp. 489–519.
  61. L. M. Leemis and J. T. McQueston, Univariate distribution relationships, Am. Stat. 62, 45 (2008).
  62. A. L. Shaw, D. K. Mark, J. Choi, R. Finkelstein, P. Scholl, S. Choi, and M. Endres, Experimental signatures of Hilbert-space ergodicity: Universal bitstring distributions and applications in noise learning, Phys. Rev. X 15, 031001 (2025).
  63. S. Mandal, P. W. Claeys, and S. Roy, Partial projected ensembles and spatiotemporal structure of information scrambling, arXiv:2508.05632.
  64. V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Math. USSR-Sbor. 1, 457 (1967).
  65. A. Milekhin and S. Murciano, Observable-projected ensembles, Quantum 9, 1888 (2025).
  66. N. D. Varikuti and S. Bandyopadhyay, Unraveling the emergence of quantum state designs in systems with symmetry, Quantum 8, 1456 (2024).
  67. A. Chan and A. De Luca, Projected state ensemble of a generic model of many-body quantum chaos, J. Phys. A 57, 405001 (2024).

Outline

Information

Phys. Rev. Lett. 135, 260402– Published 29 December, 2025
  • Received 21 May 2025
  • Accepted 9 December 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. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation