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Effects of monitoring on entanglement dynamics for (1+1)DZ2 lattice gauge theory

Nilachal Chakrabarti1,*, Nisa Ara2,3,4,†, Neha Nirbhan1,‡, Arpan Bhattacharyya1,§, and Indrakshi Raychowdhury3,4,∥

  • *Contact author: nilachalchakrabarti@iitgn.ac.in
  • †Contact author: nisa.ara@theory.tifr.res.in
  • ‡Contact author: neha.nirbhan1207@gmail.com
  • §Contact author: abhattacharyya@iitgn.ac.in
  • ∥Contact author: indrakshir@goa.bits-pilani.ac.in
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APS Open Sci. 1, 000118 – Published 1 September, 2026

DOI: https://doi.org/10.1103/1tmj-fftr

Abstract

The (1+1)-dimensional Z2 gauge theory is the simplest model that allows for quantum simulation to probe the fundamental aspects of a gauge theory coupled with dynamical fermions. To reliably benchmark such a system, it is crucial to understand the nonunitary quantum dynamics arising from effective non-Hermitian evolution and postselected monitoring protocols. This work focuses on the postselected non-Hermitian filtering dynamics of a Z2 gauge theory, where the non-Hermitian terms are associated with local and nonlocal gauge-invariant operators naturally present in the theory. We interpret the resulting dynamics as postselected filtering, where different operator sectors are coupled to loss channels with different rates. This gives a unified framework for both the local electric flux and particle-number terms and the nonlocal mesonic hopping term. Tensor network calculations are performed to probe the effect of the filtering for larger lattice sizes (up to 256-site systems). Using matrix product state calculations, the dynamics of entanglement entropy are studied as a function of the filtering rate and the coupling constant. We find that, under both local and nonlocal filtering, the late-time saturation value of the bipartite entanglement entropy remains independent of system size, providing no evidence of a measurement-induced phase transitionlike phenomenon in the postselected dynamics across the range of filtering strengths, evolution times, and system sizes considered here.

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References (102)

  1. J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51, 659 (1979).
  2. D. Banerjee, M. Dalmonte, M. Müller, E. Rico, P. Stebler, U.-J. Wiese, et al., Atomic quantum simulation of dynamical gauge fields coupled to fermionic matter: From string breaking to evolution after a quench, Phys. Rev. Lett. 109, 175302 (2012).
  3. E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices, Rep. Prog. Phys. 79, 014401 (2016).
  4. E. A. Martinez, C. A. Muschik, P. Schindler, D. Nigg, A. Erhard, M. Heyl, et al. , Real-time dynamics of lattice gauge theories with a few-qubit quantum computer, Nature (London) 534, 516 (2016).
  5. B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C. Halimeh, et al., Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator, Nature (London) 587, 392 (2020).
  6. E. Zohar, Quantum simulation of lattice gauge theories in more than one space dimension—Requirements, challenges and methods, Phil. Trans. R. Soc. A 380, 20210069 (2022).
  7. Z.-Y. Zhou, G.-X. Su, J. C. Halimeh, R. Ott, H. Sun, P. Hauke, et al., Thermalization dynamics of a gauge theory on a quantum simulator, Science 377, 311 (2022).
  8. H.-Y. Wang, W.-Y. Zhang, Z. Yao, Y. Liu, Z.-H. Zhu, Y.-G. Zheng, et al., Interrelated thermalization and quantum criticality in a lattice gauge simulator, Phys. Rev. Lett. 131, 050401 (2023).
  9. J.-Y. Desaules, G.-X. Su, I. P. McCulloch, B. Yang, Z. Papić, and J. C. Halimeh, Ergodicity breaking under confinement in cold-atom quantum simulators, Quantum 8, 1274 (2024).
  10. N. Mueller, T. Wang, O. Katz, Z. Davoudi, and M. Cetina, Quantum computing universal thermalization dynamics in a (2 + 1)D lattice gauge theory, Nat. Commun. 16, 5492 (2025).
  11. W.-Y. Zhang, Y. Liu, Y. Cheng, M.-G. He, H.-Y. Wang, T.-Y. Wang, et al., Observation of microscopic confinement dynamics by a tunable topological θ-angle, Nat. Phys. 21, 155 (2025).
  12. J. J. Osborne, I. P. McCulloch, B. Yang, P. Hauke, and J. C. Halimeh, Large-scale 2+1D U(1) gauge theory with dynamical matter in a cold-atom quantum simulator, Commun. Phys. 8, 273 (2025).
  13. M. Kang, S. Kim, Y. Qian, P. M. Neves, L. Ye, J. Jung, et al., Measurements of the quantum geometric tensor in solids, Nat. Phys. 21, 110 (2025).
  14. J. C. Halimeh, N. Mueller, J. Knolle, Z. Papić, and Z. Davoudi, Quantum simulation of out-of-equilibrium dynamics in gauge theories, arXiv:2509.03586.
  15. F. J. Wegner, Duality in generalized Ising models and phase transitions without local order parameters, J. Math. Phys. 12, 2259 (1971).
  16. E. H. Fradkin and L. Susskind, Order and disorder in gauge systems and magnets, Phys. Rev. D 17, 2637 (1978).
  17. E. Fradkin and S. H. Shenker, Phase diagrams of lattice gauge theories with Higgs fields, Phys. Rev. D 19, 3682 (1979).
  18. T. Senthil and M. P. A. Fisher, Z2 gauge theory of electron fractionalization in strongly correlated systems, Phys. Rev. B 62, 7850 (2000).
  19. H.-H. Lai and O. I. Motrunich, Majorana spin liquids on a two-leg ladder, Phys. Rev. B 84, 235148 (2011).
  20. M. Kormos, M. Collura, G. Takács, and P. Calabrese, Real-time confinement following a quantum quench to a non-integrable model, Nat. Phys. 13, 246 (2017).
  21. A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, Disorder-free localization, Phys. Rev. Lett. 118, 266601 (2017).
  22. A. Smith, J. Knolle, R. Moessner, and D. L. Kovrizhin, Dynamical localization in Z2 lattice gauge theories, Phys. Rev. B 97, 245137 (2018).
  23. J. Frank, E. Huffman, and S. Chandrasekharan, Emergence of Gauss’ law in a Z2 lattice gauge theory in 1+1 dimensions, Phys. Lett. B 806, 135484 (2020).
  24. U. Borla, R. Verresen, F. Grusdt, and S. Moroz, Confined phases of one-dimensional spinless fermions coupled to Z2 gauge theory, Phys. Rev. Lett. 124, 120503 (2020).
  25. E. Zohar, A. Farace, B. Reznik, and J. I. Cirac, Digital quantum simulation of Z2 lattice gauge theories with dynamical fermionic matter, Phys. Rev. Lett. 118, 070501 (2017).
  26. F. Grusdt and L. Pollet, Z2 parton phases in the mixed-dimensional t−Jz model, Phys. Rev. Lett. 125, 256401 (2020).
  27. C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman, et al., Floquet approach to Z2 lattice gauge theories with ultracold atoms in optical lattices, Nat. Phys. 15, 1168 (2019).
  28. F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and M. Dalmonte, Lattice gauge theories and string dynamics in Rydberg atom quantum simulators, Phys. Rev. X 10, 021041 (2020).
  29. M. C. Bañuls et al., Simulating lattice gauge theories within quantum technologies, Eur. Phys. J. D 74, 165 (2020).
  30. U. Borla, R. Verresen, J. Shah, and S. Moroz, Gauging the Kitaev chain, SciPost Phys. 10, 148 (2021).
  31. J. C. Halimeh, L. Homeier, C. Schweizer, M. Aidelsburger, P. Hauke, and F. Grusdt, Stabilizing lattice gauge theories through simplified local pseudogenerators, Phys. Rev. Res. 4, 033120 (2022).
  32. J. Mildenberger, W. Mruczkiewicz, J. C. Halimeh, Z. Jiang, and P. Hauke, Confinement in a Z2 lattice gauge theory on a quantum computer, Nat. Phys. 21, 312 (2025).
  33. Z. Davoudi, N. Mueller, and C. Powers, Towards quantum computing phase diagrams of gauge theories with thermal pure quantum states, Phys. Rev. Lett. 131, 081901 (2023).
  34. M. Kebrič, U. Schollwöck, and F. Grusdt, Mean-field theory of 1+1D Z2 lattice gauge theory with matter, SciPost Phys. 20, 017 (2026).
  35. I.-C. Chen, J. C. Getelina, K. Pollock, A. Khindanov, S. Sen, Y.-X. Yao, et al., Classical and quantum simulations of 1+1-dimensional Z2 gauge theory at finite temperature and density, Commun. Phys. 8, 375 (2025).
  36. C. Alexandrou, A. Athenodorou, K. Blekos, G. Polykratis, and S. Kühn, Realizing string breaking dynamics in a Z2 lattice gauge theory on quantum hardware, Phys. Rev. D 112, 114506 (2025).
  37. T. Pichler, E. Rico, M. Dalmonte, S. Montangero, and P. Zoller, Real-time dynamics in lattice gauge theories with tensor networks, Phys. Rev. X 6, 011023 (2016).
  38. J. Knaute, M. Feuerstein, and E. Zohar, Entanglement and confinement in lattice gauge theory tensor networks, J. High Energy Phys. 02 (2024) 174.
  39. M. Fromm, O. Philipsen, M. Spannowsky, and C. Winterowd, Simulating Z2 lattice gauge theory with the variational quantum thermalizer, EPJ Quantum Technol. 11, 20 (2024).
  40. H. Casini, M. Huerta, and J. A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89, 085012 (2014).
  41. M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and S. Kühn, Tensor networks and their use for lattice gauge theories, PoS LATTICE2018, 022 (2018).
  42. G. Magnifico, M. Dalmonte, P. Facchi, S. Pascazio, F. V. Pepe, and E. Ercolessi, Real time dynamics and confinement in the Zn Schwinger-Weyl lattice model for 1+1 QED, Quantum 4, 281 (2020).
  43. R. Irmejs, M. C. Banuls, and J. I. Cirac, Quantum simulation of Z2 lattice gauge theory with minimal resources, Phys. Rev. D 108, 074503 (2023).
  44. E. Mathew, N. Gupta, S. V. Kadam, A. Bapat, J. Stryker, Z. Davoudi, and I. Raychowdhury, Tensor-network toolbox for probing dynamics of non-Abelian gauge theories, PoS LATTICE2024, 472 (2025).
  45. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  46. Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  47. Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  48. A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019).
  49. T. Boorman, M. Szyniszewski, H. Schomerus, and A. Romito, Diagnostics of entanglement dynamics in noisy and disordered spin chains via the measurement-induced steady-state entanglement transition, Phys. Rev. B 105, 144202 (2022).
  50. A. Biella and M. Schiró, Many-body quantum Zeno effect and measurement-induced subradiance transition, Quantum 5, 528 (2021).
  51. M. Szyniszewski, A. Romito, and H. Schomerus, Universality of entanglement transitions from stroboscopic to continuous measurements, Phys. Rev. Lett. 125, 210602 (2020).
  52. F. Barratt, U. Agrawal, A. C. Potter, S. Gopalakrishnan, and R. Vasseur, Transitions in the learnability of global charges from local measurements, Phys. Rev. Lett. 129, 200602 (2022).
  53. A. Zabalo, J. H. Wilson, M. J. Gullans, R. Vasseur, S. Gopalakrishnan, D. A. Huse, et al., Infinite-randomness criticality in monitored quantum dynamics with static disorder, Phys. Rev. B 107, L220204 (2023).
  54. F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter, Field theory of charge sharpening in symmetric monitored quantum circuits, Phys. Rev. Lett. 129, 120604 (2022).
  55. O. Lunt and A. Pal, Measurement-induced entanglement transitions in many-body localized systems, Phys. Rev. Res. 2, 043072 (2020).
  56. A. Zabalo, M. J. Gullans, J. H. Wilson, R. Vasseur, A. W. W. Ludwig, S. Gopalakrishnan, et al., Operator scaling dimensions and multifractality at measurement-induced transitions, Phys. Rev. Lett. 128, 050602 (2022).
  57. J. Iaconis and X. Chen, Multifractality in nonunitary random dynamics, Phys. Rev. B 104, 214307 (2021).
  58. P. Sierant, G. Chiriacò, F. M. Surace, S. Sharma, X. Turkeshi, M. Dalmonte, et al., Dissipative Floquet dynamics: From steady state to measurement induced criticality in trapped-ion chains, Quantum 6, 638 (2022).
  59. Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020).
  60. S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum error correction in scrambling dynamics and measurement-induced phase transition, Phys. Rev. Lett. 125, 030505 (2020).
  61. M. Szyniszewski, A. Romito, and H. Schomerus, Entanglement transition from variable-strength weak measurements, Phys. Rev. B 100, 064204 (2019).
  62. M. Block, Y. Bao, S. Choi, E. Altman, and N. Y. Yao, Measurement-induced transition in long-range interacting quantum circuits, Phys. Rev. Lett. 128, 010604 (2022).
  63. C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum circuits, Phys. Rev. B 101, 104302 (2020).
  64. U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, et al., Entanglement and charge-sharpening transitions in U(1) symmetric monitored quantum circuits, Phys. Rev. X 12, 041002 (2022).
  65. M. J. Gullans and D. A. Huse, Scalable probes of measurement-induced criticality, Phys. Rev. Lett. 125, 070606 (2020).
  66. S. Sharma, X. Turkeshi, R. Fazio, and M. Dalmonte, Measurement-induced criticality in extended and long-range unitary circuits, SciPost Phys. Core 5, 023 (2022).
  67. A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrishnan, D. A. Huse, and J. H. Pixley, Critical properties of the measurement-induced transition in random quantum circuits, Phys. Rev. B 101, 060301 (2020).
  68. R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. W. Ludwig, Entanglement transitions from holographic random tensor networks, Phys. Rev. B 100, 134203 (2019).
  69. Y. Li, X. Chen, A. W. W. Ludwig, and M. P. A. Fisher, Conformal invariance and quantum nonlocality in critical hybrid circuits, Phys. Rev. B 104, 104305 (2021).
  70. X. Turkeshi, R. Fazio, and M. Dalmonte, Measurement-induced criticality in (2+1)-dimensional hybrid quantum circuits, Phys. Rev. B 102, 014315 (2020).
  71. P. Sierant and X. Turkeshi, Universal behavior beyond multifractality of wave functions at measurement-induced phase transitions, Phys. Rev. Lett. 128, 130605 (2022).
  72. M. J. Gullans and D. A. Huse, Dynamical purification phase transition induced by quantum measurements, Phys. Rev. X 10, 041020 (2020).
  73. K. Yamamoto and R. Hamazaki, Localization properties in disordered quantum many-body dynamics under continuous measurement, Phys. Rev. B 107, L220201 (2023).
  74. Y. Han and X. Chen, Measurement-induced criticality in Z2-symmetric quantum automaton circuits, Phys. Rev. B 105, 064306 (2022).
  75. M. M. Wauters, E. Ballini, A. Biella, and P. Hauke, Symmetry-protection Zeno phase transition in monitored lattice gauge theories, Phys. Rev. B 111, 094315 (2025).
  76. Y. Li, D. S. Bhakuni, Y.-C. Liu, and M. Dalmonte, Metastable confinement in Rydberg lattice gauge theories, arXiv:2602.22890.
  77. R. Maeno, Gauge-invariant QMETTS with mutually unbiased physical bases for Z2 lattice gauge theories at finite temperature and density, arXiv:2603.10932.
  78. M. L. Rhodes, S. Pathak, and R. W. Chien, Quantum simulation of lattice gauge theories coupled to fermionic matter via anyonic regularization, arXiv:2603.15820.
  79. J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992).
  80. H. J. Carmichael, Quantum trajectory theory for cascaded open systems, Phys. Rev. Lett. 70, 2273 (1993).
  81. M. B. Plenio and P. L. Knight, The quantum jump approach to dissipative dynamics in quantum optics, Rev. Mod. Phys. 70, 101 (1998).
  82. X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schiró, Measurement-induced entanglement transitions in the quantum Ising chain: From infinite to zero clicks, Phys. Rev. B 103, 224210 (2021).
  83. Y. L. Gal, X. Turkeshi, and M. Schirò, Volume-to-area law entanglement transition in a non-Hermitian free fermionic chain, SciPost Phys. 14, 138 (2023).
  84. Z. Davoudi, C.-C. Hsieh, and S. V. Kadam, Scattering wave packets of hadrons in gauge theories: Preparation on a quantum computer, Quantum 8, 1520 (2024).
  85. N. Ara, A. Bhattacharyya, N. Chakrabarti, N. Nirbhan, and I. Raychowdhury, Dynamics of entanglement entropy for a locally monitored lattice gauge theory, PoS LATTICE 2025, 136 (2026).
  86. H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, Germany, 1993).
  87. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2009).
  88. C. Gardiner and P. Zoller, Quantum Noise (Springer Science & Business Media, 2004).
  89. A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
  90. K. Jacobs, Quantum Measurement Theory and its Applications (Cambridge University Press, 2014).
  91. K. Yamamoto, M. Nakagawa, M. Tezuka, M. Ueda, and N. Kawakami, Universal properties of dissipative Tomonaga-Luttinger liquids: Case study of a non-Hermitian XXZ spin chain, Phys. Rev. B 105, 205125 (2022).
  92. C. Zerba and A. Silva, Measurement phase transitions in the no-click limit as quantum phase transitions of a non-Hermitean vacuum, SciPost Phys. Core 6, 051 (2023).
  93. N. Chakrabarti, N. Nirbhan, and A. Bhattacharyya, Dynamics of monitored SSH model in Krylov space: From complexity to quantum Fisher information, J. High Energy Phys. 07 (2025) 203.
  94. R. Orus, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  95. J. C. Bridgeman and C. T. Chubb, Hand-waving and interpretive dance: An introductory course on tensor networks, J. Phys. A: Math. Theor. 50, 223001 (2017).
  96. M. Fishman, S. R. White, and E. M. Stoudenmire, Codebase release 0.3 for ITensor, SciPost Phys. Codebases 4-r0.3 (2022).
  97. X. Yang, X.-P. Jiang, and L. Pan, Dissipation-enhanced localization in a disorder-free Z2 lattice gauge system, arXiv:2509.06642.
  98. M. Yang and S. R. White, Time-dependent variational principle with ancillary Krylov subspace, Phys. Rev. B 102, 094315 (2020).
  99. P. Facchi, H. Nakazato, and S. Pascazio, From the quantum Zeno to the inverse quantum Zeno effect, Phys. Rev. Lett. 86, 2699 (2001).
  100. J. Bringewatt, J. Kunjummen, and N. Mueller, Randomized measurement protocols for lattice gauge theories, Quantum 8, 1300 (2024).
  101. I. Raychowdhury and J. R. Stryker, Loop, string, and hadron dynamics in SU(2) Hamiltonian lattice gauge theories, Phys. Rev. D 101, 114502 (2020).
  102. N. Chakrabarti, Effects of monitoring in Z2 gauge theory, GitHub, 2026, https://github.com/Nilachal-QIQMB/Effects-of-Monitoring-in-Z2-gauge-theory-.git.

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APS Open Sci. 1, 000118– Published 1 September, 2026
Vol. 1
  • Received 7 May 2026
  • Revised 10 July 2026
  • Accepted 31 July 2026
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