- 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

Robust Detection of an Entanglement Transition in the Projective Transverse-Field Ising Model

Felix Roser*, Etienne M. Springer, Hans Peter Büchler, and Nicolai Lang

  • *Contact author: felix.roser@itp3.uni-stuttgart.de
PDF

Phys. Rev. Lett. 136, 140403 – Published 10 April, 2026

DOI: https://doi.org/10.1103/5m5q-f7vc

Abstract

We propose a scalable and noise-resilient protocol for the detection of the entanglement transition in a projective version of the transverse-field Ising model. Entanglement transitions are experimentally difficult to observe due to the inherent randomness of projective measurements and noise in large-scale experimental settings. Our approach combines error correction algorithms with classical shadow tomography to overcome both problems. This allows for experimentally accessible upper and lower bounds on the entanglement transition without postselection or full state tomography. These bounds remain robust under noise and their sharpness is a measure of the noise rate.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (50)

  1. Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  2. Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  3. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  4. A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, in Entanglement in Spin Chains: From Theory to Quantum Technology Applications, edited by A. Bayat, S. Bose, and H. Johannesson (Springer International Publishing, Cham, 2022), pp. 211–249.
  5. M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annu. Rev. Condens. Matter Phys. 14, 335 (2023).
  6. N. Lang and H. P. Büchler, Entanglement transition in the projective transverse field Ising model, Phys. Rev. B 102, 094204 (2020).
  7. A. Nahum and B. Skinner, Entanglement and dynamics of diffusion-annihilation processes with Majorana defects, Phys. Rev. Res. 2, 023288 (2020).
  8. M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, Entanglement phase transitions in measurement-only dynamics, Phys. Rev. X 11, 011030 (2021).
  9. J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer, New York, 1932), 10.1007/978-3-642-61409-5.
  10. C. Noel, P. Niroula, D. Zhu, A. Risinger, L. Egan, D. Biswas, M. Cetina, A. V. Gorshkov, M. J. Gullans, D. A. Huse, and C. Monroe, Measurement-induced quantum phases realized in a trapped-ion quantum computer, Nat. Phys. 18, 760 (2022).
  11. J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout, Nat. Phys. 19, 1314 (2023).
  12. J. C. Hoke et al. (Collaborators), Measurement-induced entanglement and teleportation on a noisy quantum processor, Nature (London) 622, 481 (2023).
  13. S. Czischek, G. Torlai, S. Ray, R. Islam, and R. G. Melko, Simulating a measurement-induced phase transition for trapped-ion circuits, Phys. Rev. A 104, 062405 (2021).
  14. S. J. Garratt and E. Altman, Probing postmeasurement entanglement without postselection, PRX Quantum 5, 030311 (2024).
  15. F. Roser, H. P. Büchler, and N. Lang, Decoding the projective transverse field Ising model, Phys. Rev. B 107, 214201 (2023).
  16. M. Ippoliti and V. Khemani, Postselection-free entanglement dynamics via spacetime duality, Phys. Rev. Lett. 126, 060501 (2021).
  17. M. Ippoliti, T. Rakovszky, and V. Khemani, Fractal, logarithmic, and volume-law entangled nonthermal steady states via spacetime duality, Phys. Rev. X 12, 011045 (2022).
  18. S. Roy, J. T. Chalker, I. V. Gornyi, and Y. Gefen, Measurement-induced steering of quantum systems, Phys. Rev. Res. 2, 033347 (2020).
  19. K. Pöyhönen, A. G. Moghaddam, M. N. Ivaki, and T. Ojanen, Postselection-free approach to monitored quantum dynamics and entanglement phase transitions, arXiv:2406.19052.
  20. Y. Li, Y. Zou, P. Glorioso, E. Altman, and M. P. A. Fisher, Cross entropy benchmark for measurement-induced phase transitions, Phys. Rev. Lett. 130, 220404 (2023).
  21. M. Tikhanovskaya, A. Lavasani, M. P. A. Fisher, and S. Vijay, Universality of the cross-entropy in Z2 symmetric monitored quantum circuits, Phys. Rev. B 109, 224313 (2024).
  22. H. Kamakari, J. Sun, Y. Li, J. J. Thio, T. P. Gujarati, M. P. A. Fisher, M. Motta, and A. J. Minnich, Experimental demonstration of scalable cross-entropy benchmarking to detect measurement-induced phase transitions on a superconducting quantum processor, Phys. Rev. Lett. 134, 120401 (2025).
  23. G. Passarelli, X. Turkeshi, A. Russomanno, P. Lucignano, M. Schirò, and R. Fazio, Many-body dynamics in monitored atomic gases without postselection barrier, Phys. Rev. Lett. 132, 163401 (2024).
  24. X. Feng, B. Skinner, and A. Nahum, Measurement-induced phase transitions on dynamical quantum trees, PRX Quantum 4, 030333 (2023).
  25. X. Feng, J. Côté, S. Kourtis, and B. Skinner, Postselection-free experimental observation of the measurement-induced phase transition in circuits with universal gates, arXiv:2502.01735.
  26. M. J. Gullans and D. A. Huse, Scalable probes of measurement-induced criticality, Phys. Rev. Lett. 125, 070606 (2020).
  27. M. J. Gullans and D. A. Huse, Dynamical purification phase transition induced by quantum measurements, Phys. Rev. X 10, 041020 (2020).
  28. M. Ippoliti and V. Khemani, Learnability transitions in monitored quantum dynamics via eavesdropper’s classical shadows, PRX Quantum 5, 020304 (2024).
  29. H. Dehghani, A. Lavasani, M. Hafezi, and M. J. Gullans, Neural-network decoders for measurement induced phase transitions, Nat. Commun. 14, 2918 (2023).
  30. A. Khindanov, I. L. Aleiner, L. Faoro, and L. B. Ioffe, Observable measurement-induced transitions, Ann. Phys. (Amsterdam) 479, 170047 (2025).
  31. S. Aaronson, Shadow tomography of quantum states, SIAM J. Comput. 49, STOC18-368 (2020).
  32. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  33. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
  34. M. McGinley, Postselection-free learning of measurement-induced quantum dynamics, PRX Quantum 5, 020347 (2024).
  35. D. Gottesman, Class of quantum error-correcting codes saturating the quantum hamming bound, Phys. Rev. A 54, 1862 (1996).
  36. D. Gottesman, The Heisenberg representation of quantum computers, arXiv:quant-ph/9807006 (1998).
  37. D. Gottesman, Theory of fault-tolerant quantum computation, Phys. Rev. A 57, 127 (1998).
  38. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  39. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010), 10.1017/CBO9780511976667.
  40. R. O’Donnell and J. Wright, Efficient quantum tomography, in Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’16 (Association for Computing Machinery, New York, NY, USA, 2016), pp. 899–912, 10.1145/2897518.2897544.
  41. T. Scharnhorst, J. Spilecki, and J. Wright, Optimal lower bounds for quantum state tomography, arXiv:2510.07699.

  42. Note that X1X2⋯XL is a symmetry of the PTIM dynamics; hence an encoding in symmetry eigenstates |Ψ0⟩∝|0…0⟩+(−1)m|1…1⟩ cannot be used to detect the entanglement transition.

  43. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5m5q-f7vc for details on error correction and state prediction for the PTIM, which includes Refs. [44,45].
  44. G. Duclos-Cianci and D. Poulin, Fast decoders for topological quantum codes, Phys. Rev. Lett. 104, 050504 (2010).
  45. A. G. Fowler, A. C. Whiteside, and L. C. L. Hollenberg, Towards practical classical processing for the surface code, Phys. Rev. Lett. 108, 180501 (2012).
  46. J. Edmonds, Paths, trees, and flowers, Can. J. Math. 17, 449 (1965).
  47. V. Kolmogorov, Blossom V: A new implementation of a minimum cost perfect matching algorithm, Math. Program. Comput. 1, 43 (2009).
  48. S. Bravyi, M. Suchara, and A. Vargo, Efficient algorithms for maximum likelihood decoding in the surface code, Phys. Rev. A 90, 032326 (2014).
  49. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, J. Math. Phys. (N.Y.) 43, 4452 (2002).
  50. F. Roser, E. M. Springer, H. P. Büchler, and N. Lang, Replication data for: “Robust detection of an entanglement transition in the projective transverse-field Ising model,” 10.18419/DARUS-5305.

Outline

Information

Phys. Rev. Lett. 136, 140403– Published 10 April, 2026
  • Received 3 December 2025
  • Revised 6 February 2026
  • Accepted 19 March 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