- Physical Review Journals
All JournalsPhysics Magazine

PRX Quantum

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

Generalized Zeno Effect and Entanglement Dynamics Induced by Fermion Counting

Elias Starchl1, Mark H. Fischer2, and Lukas M. Sieberer1,*

  • *Contact author: lukas.sieberer@uibk.ac.at
PDF

PRX Quantum 6, 030302 – Published 2 July, 2025

DOI: https://doi.org/10.1103/jppz-vdgn

Abstract

We study a one-dimensional lattice system of free fermions subjected to a generalized measurement process: the system exchanges particles with its environment, but each fermion leaving or entering the system is counted. In contrast to the freezing of dynamics due to frequent measurements of lattice site occupation numbers, a high rate of fermion counts induces fast fluctuations in the state of the system. Still, through numerical simulations of quantum trajectories and an analytical approach based on replica Keldysh field theory, we find that instantaneous correlations and entanglement properties of free fermions subjected to fermion counting and local occupation measurements are strikingly similar. We explain this similarity through a generalized Zeno effect induced by fermion counting and a universal long-wavelength description in terms of a nonlinear sigma model. The physical requirements underlying this universal emergent behavior are conservation of the total number of particles in the system and its environment, and conservation of the purity of the state of the system by keeping a full record of all measurement outcomes. For both types of measurement processes, we present strong evidence against the existence of a critical phase with logarithmic entanglement and conformal invariance. Instead, we identify a finite critical range of length scales on which signatures of conformal invariance are observable. While area-law entanglement is established beyond a scale that is exponentially large in the measurement rate, the upper boundary of the critical range is only algebraically large and thus numerically accessible. Our finding that these properties do not rely on particle-number conservation has far-reaching implications for measurement-induced phenomena beyond noninteracting fermions, such as charge sharpening in random quantum circuits or generic interacting systems.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Unitary dynamics leads to the buildup of entanglement between different spatial regions of a quantum many-body system. In contrast, local measurements decouple the regions by destroying the entanglement between the measured region and the rest of the system. The competition between unitary dynamics and repeated local measurements can lead to a phase transition characterized by a qualitative change in the entanglement dynamics of the many-body wave function. In our work, we clarify the impact of particle-number conservation on such measurement-induced entanglement dynamics of a one-dimensional free-fermionic lattice system.

Specifically, we compare two types of generalized measurement processes: monitored loss and gain of particles and measurements of the occupations of lattice sites. While the particle number in the system is conserved for occupation measurements, implementing monitored loss and gain requires coupling the system to particle reservoirs, such that only the total particle number in the system and reservoirs is conserved. Surprisingly, we find that both types of measurements result in almost identical entanglement properties. Using a field-theoretical treatment, we trace this similarity back to a symmetry common to both models. We confirm the absence of an entanglement transition for either measurement protocol, contrary to Majorana models, where particle-number conservation is violated.

Our results have far-reaching implications beyond noninteracting fermions. We expect that measurement-induced phenomena that have been observed in systems with a conservation law such as charge sharpening in random quantum circuits or generic interacting fermionic systems also occur under the much weaker condition of global charge conservation.

Article Text

References (128)

  1. B. Misra and E. C. G. Sudarshan, The Zeno’s paradox in quantum theory, J. Math. Phys. 18, 756 (1977).
  2. Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  3. B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  4. M. Szyniszewski, Entanglement transition from variable-strength weak measurements, Phys. Rev. B 100, 064204 (2019).
  5. Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  6. Y. Bao, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020).
  7. M. J. Gullans and D. A. Huse, Dynamical purification phase transition induced by quantum measurements, Phys. Rev. X 10, 041020 (2020).
  8. X. Turkeshi, R. Fazio, and M. Dalmonte, Measurement-induced criticality in (2+1)-dimensional hybrid quantum circuits, Phys. Rev. B 102, 014315 (2020).
  9. 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).
  10. Y. Bao, S. Choi, and E. Altman, Symmetry enriched phases of quantum circuits, Ann. Phys. (N. Y). 435, 168618 (2021).
  11. 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).
  12. A. Nahum, Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory, PRX Quantum 2, 010352 (2021).
  13. A. Lavasani, Y. Alavirad, and M. Barkeshli, Measurement-induced topological entanglement transitions in symmetric random quantum circuits, Nat. Phys. 17, 342 (2021).
  14. P. Sierant, M. Schirò, M. Lewenstein, and X. Turkeshi, Measurement-induced phase transitions in (d+1)-dimensional stabilizer circuits, Phys. Rev. B 106, 214316 (2022).
  15. A. Zabalo, M. J. Gullans, J. H. Wilson, R. Vasseur, A. W. W. Ludwig, S. Gopalakrishnan, D. A. Huse, and J. H. Pixley, Operator scaling dimensions and multifractality at measurement-induced transitions, Phys. Rev. Lett. 128, 050602 (2022).
  16. X. Yu and X.-L. Qi, Measurement-induced entanglement phase transition in random bilocal circuits, arXiv:2201.12704.
  17. U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening transitions in U(1) symmetric monitored quantum circuits, Phys. Rev. X 12, 41002 (2022).
  18. 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).
  19. C.-M. Jian, B. Bauer, A. Keselman, and A. W. W. Ludwig, Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions, Phys. Rev. B 106, 134206 (2022).
  20. C.-M. Jian, H. Shapourian, B. Bauer, and A. W. W. Ludwig, Measurement-induced entanglement transitions in quantum circuits of non-interacting fermions: Born-rule versus forced measurements, arXiv:2302.09094.
  21. H. Oshima and Y. Fuji, Charge fluctuation and charge-resolved entanglement in a monitored quantum circuit with U(1) symmetry, Phys. Rev. B 107, 014308 (2023).
  22. S. P. Kelly, U. Poschinger, F. Schmidt-Kaler, M. P. A. Fisher, and J. Marino, Coherence requirements for quantum communication from hybrid circuit dynamics, SciPost Physics 15, 250 (2023).
  23. Z. Weinstein, S. P. Kelly, J. Marino, and E. Altman, Scrambling transition in a radiative random unitary circuit, Phys. Rev. Lett. 131, 220404 (2023).
  24. S. P. Kelly and J. Marino, Entanglement transitions induced by quantum-data collection, Phys. Rev. A 111, L010402 (2025).
  25. S. P. Kelly and J. Marino, Generalizing measurement-induced phase transitions to information exchange symmetry breaking, Phys. Rev. A 111, 012425 (2025).
  26. X. Cao, A. Tilloy, and A. De Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys. 7, 024 (2019).
  27. X. Chen, Emergent conformal symmetry in nonunitary random dynamics of free fermions, Phys. Rev. Res. 2, 033017 (2020).
  28. A. Nahum, Entanglement and dynamics of diffusion-annihilation processes with Majorana defects, Phys. Rev. Res. 2, 023288 (2020).
  29. O. Alberton, M. Buchhold, and S. Diehl, Entanglement transition in a monitored free-fermion chain: from extended criticality to area law, Phys. Rev. Lett. 126, 170602 (2021).
  30. Q. Tang, Quantum criticality in the nonunitary dynamics of (2+1)-dimensional free fermions, Phys. Rev. B 103, 174303 (2021).
  31. M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl, Effective theory for the measurement-induced phase transition of Dirac fermions, Phys. Rev. X 11, 041004 (2021).
  32. M. Van Regemortel, Z.-P. Cian, A. Seif, H. Dehghani, and M. Hafezi, Entanglement entropy scaling transition under competing monitoring protocols, Phys. Rev. Lett. 126, 123604 (2021).
  33. M. Coppola, E. Tirrito, D. Karevski, and M. Collura, Growth of entanglement entropy under local projective measurements, Phys. Rev. B 105, 094303 (2022).
  34. F. Carollo and V. Alba, Entangled multiplets and unusual spreading of quantum correlations in a continuously monitored tight-binding chain, Phys. Rev. B 106, L220304 (2022).
  35. B. Ladewig, S. Diehl, and M. Buchhold, Monitored open fermion dynamics: Exploring the interplay of measurement, decoherence, and free Hamiltonian evolution, Phys. Rev. Res. 4, 033001 (2022).
  36. M. Buchhold, T. Müller, and S. Diehl, Revealing measurement-induced phase transitions by pre-selection, arXiv:2208.10506.
  37. X. Turkeshi, Enhanced entanglement negativity in boundary-driven monitored fermionic chains, Phys. Rev. B 106, 024304 (2022).
  38. Q. Yang, Y. Zuo, and D. E. Liu, Keldysh nonlinear sigma model for a free-fermion gas under continuous measurements, Phys. Rev. Res. 5, 033174 (2023).
  39. M. Szyniszewski, O. Lunt, and A. Pal, Disordered monitored free fermions, Phys. Rev. B 108, 165126 (2023).
  40. H. Lóio, A. De Luca, J. De Nardis, and X. Turkeshi, Purification timescales in monitored fermions, Phys. Rev. B 108, L020306 (2023).
  41. Y. Le Gal, X. Turkeshi, and M. Schirò, Volume-to-area law entanglement transition in a non-Hermitian free fermionic chain, SciPost Phys. 14, 138 (2023).
  42. I. Poboiko, P. Pöpperl, I. V. Gornyi, and A. D. Mirlin, Theory of free fermions under random projective measurements, Phys. Rev. X 13, 041046 (2023).
  43. I. Poboiko, I. V. Gornyi, and A. D. Mirlin, Measurement-induced phase transition for free fermions above one dimension, Phys. Rev. Lett. 132, 110403 (2024).
  44. G. Kells, D. Meidan, and A. Romito, Topological transitions in weakly monitored free fermions, SciPost Phys. 14, 031 (2023).
  45. M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum, Nonlinear sigma models for monitored dynamics of free fermions, Phys. Rev. X 13, 041045 (2023).
  46. J. Merritt, Entanglement transitions with free fermions, Phys. Rev. B 107, 064303 (2023).
  47. K. Klocke, Majorana loop models for measurement-only quantum circuits, Phys. Rev. X 13, 041028 (2023).
  48. K. Chahine and M. Buchhold, Entanglement phases, localization, and multifractality of monitored free fermions in two dimensions, Phys. Rev. B 110, 054313 (2024).
  49. M. Eissler, I. Lesanovsky, and F. Carollo, Unraveling-induced entanglement phase transition in diffusive trajectories of continuously monitored noninteracting fermionic systems, Phys. Rev. A 111, 022205 (2025).
  50. L. Lumia, E. Tirrito, R. Fazio, and M. Collura, Measurement-induced transitions beyond Gaussianity: A single particle description, Phys. Rev. Res. 6, 023176 (2024).
  51. M. Tsitsishvili, D. Poletti, M. Dalmonte, and G. Chiriacò, Measurement induced transitions in non-Markovian free fermion ladders, SciPost Phys. Core 7, 011 (2024).
  52. M. Fava, L. Piroli, D. Bernard, and A. Nahum, Monitored fermions with conserved U(1) charge, Phys. Rev. Res. 6, 043246 (2024).
  53. H. Guo, M. S. Foster, C.-M. Jian, and A. W. W. Ludwig, Field theory of monitored, interacting fermion dynamics with charge conservation, arXiv:2410.07317.
  54. I. Poboiko, P. Pöpperl, I. V. Gornyi, and A. D. Mirlin, Measurement-induced transitions for interacting fermions, Phys. Rev. B 111, 024204 (2025).
  55. Y. Fuji, Measurement-induced quantum criticality under continuous monitoring, Phys. Rev. B 102, 054302 (2020).
  56. S. Goto and I. Danshita, Measurement-induced transitions of the entanglement scaling law in ultracold gases with controllable dissipation, Phys. Rev. A 102, 033316 (2020).
  57. Q. Tang, Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations, Phys. Rev. Research 2, 013022 (2020).
  58. E. V. H. Doggen, Generalized quantum measurements with matrix product states: Entanglement phase transition and clusterization, Phys. Rev. Res. 4, 023146 (2022).
  59. Y. Minoguchi, P. Rabl, and M. Buchhold, Continuous Gaussian measurements of the free boson CFT: A model for exactly solvable and detectable measurement-induced dynamics, SciPost Phys. 12, 009 (2022).
  60. N. Lang, Entanglement transition in the projective transverse field Ising model, Phys. Rev. B 102, 094204 (2020).
  61. D. Rossini, Measurement-induced dynamics of many-body systems at quantum criticality, Phys. Rev. B 102, 035119 (2020).
  62. T. Botzung, S. Diehl, and M. Müller, Engineered dissipation induced entanglement transition in quantum spin chains: from logarithmic growth to area law, Phys. Rev. B 104, 184422 (2021).
  63. X. Turkeshi, Measurement-induced entanglement transitions in the quantum Ising chain: from infinite to zero clicks, Phys. Rev. B 103, 224210 (2021).
  64. Z. Weinstein, Nonlocality and entanglement in measured critical quantum Ising chains, Phys. Rev. B 107, 245132 (2023).
  65. Z. Yang, Entanglement in a one-dimensional critical state after measurements, Phys. Rev. B 108, 165120 (2023).
  66. E. Tirrito, A. Santini, R. Fazio, and M. Collura, Full counting statistics as probe of measurement-induced transitions in the quantum Ising chain, SciPost Phys. 15, 096 (2023).
  67. 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).
  68. U. Agrawal, J. Lopez-Piqueres, R. Vasseur, S. Gopalakrishnan, and A. C. Potter, Observing quantum measurement collapse as a learnability phase transition, Phys. Rev. X 14, 041012 (2024).
  69. Google Quantum AI and Collaborators, Measurement-induced entanglement and teleportation on a noisy quantum processor, Nature 622, 481 (2023).
  70. 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).
  71. H. Kamakari, J. Sun, Y. Li, J. J. Thio, T. P. Gujarati, M. P. 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).
  72. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2012), 10th ed.
  73. C. Gardiner and P. Zoller, The Quantum World of Ultra-Cold Atoms and Light Book I: Foundations of Quantum Optics (Imperial College Press, London, 2014), Vol. 2.
  74. C. Gardiner and P. Zoller, The Quantum World of Ultra-Cold Atoms and Light Book II: The Physics of Quantum-Optical Devices (Imperial College Press, London, 2015), Vol. 4.
  75. K. Jacobs and D. A. Steck, A straightforward introduction to continuous quantum measurement, Contemp. Phys. 47, 279 (2006).
  76. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2009), 1st ed.
  77. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics (Springer New York, New York, NY, 1997).
  78. C. Gardiner, Stochastic Methods (Springer Berlin, Heidelberg, Berlin, Heidelberg, 2009), 4th ed.
  79. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A Math. Theor. 42, 504005 (2009).
  80. A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
  81. S. Gustavsson, R. Leturcq, M. Studer, I. Shorubalko, T. Ihn, K. Ensslin, D. C. Driscoll, and A. C. Gossard, Electron counting in quantum dots, Surf. Sci. Rep. 64, 191 (2009).
  82. E. Starchl and L. M. Sieberer, Relaxation to a parity-time symmetric generalized Gibbs ensemble after a quantum quench in a driven-dissipative Kitaev chain, Phys. Rev. Lett. 129, 220602 (2022).
  83. E. Starchl and L. M. Sieberer, Quantum quenches in driven-dissipative quadratic fermionic systems with parity-time symmetry, Phys. Rev. Res. 6, 013016 (2024).
  84. D. S. Abrams and S. Lloyd, Simulation of many-body fermi systems on a universal quantum computer, Phys. Rev. Lett. 79, 2586 (1997).
  85. G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Quantum algorithms for fermionic simulations, Phys. Rev. A 64, 022319 (2001).
  86. S. B. Bravyi and A. Y. Kitaev, Fermionic quantum computation, Ann. Phys. (N. Y). 298, 210 (2002).
  87. R. C. Ball, Fermions without fermion fields, Phys. Rev. Lett. 95, 176407 (2005).
  88. F. Verstraete and J. I. Cirac, Mapping local Hamiltonians of fermions to local Hamiltonians of spins, J. Stat. Mech. Theory Exp. 2005, P09012 (2005).
  89. J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Simulation of electronic structure Hamiltonians using quantum computers, Mol. Phys. 109, 735 (2011).
  90. R. Barends et al., Digital quantum simulation of fermionic models with a superconducting circuit, Nat. Commun. 6, 7654 (2015).
  91. J. D. Whitfield, V. Havlíček, and M. Troyer, Local spin operators for fermion simulations, Phys. Rev. A 94, 030301 (2016).
  92. D. González-Cuadra, D. Bluvstein, M. Kalinowski, R. Kaubruegger, N. Maskara, P. Naldesi, T. V. Zache, A. M. Kaufman, M. D. Lukin, H. Pichler, B. Vermersch, J. Ye, and P. Zoller, Fermionic quantum processing with programmable neutral atom arrays, Proc. Natl. Acad. Sci. 120, 1 (2023).
  93. D. Bernard, T. Jin, and O. Shpielberg, Transport in quantum chains under strong monitoring, EPL (Europhysics Lett.) 121, 60006 (2018).
  94. T. Chou, K. Mallick, and R. K. P. Zia, Non-equilibrium statistical mechanics: from a paradigmatic model to biological transport, Rep. Prog. Phys. 74, 116601 (2011).
  95. K. Mallick, The exclusion process: A paradigm for non-equilibrium behaviour, Phys. A Stat. Mech. Appl. 418, 17 (2015).
  96. D. T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem. 58, 35 (2007), pMID: 17037977.
  97. G. L. Daquila and U. C. Täuber, Slow relaxation and aging kinetics for the driven lattice gas, Phys. Rev. E 83, 051107 (2011).
  98. I. Klich and L. Levitov, Quantum noise as an entanglement meter, Phys. Rev. Lett. 102, 100502 (2009).
  99. H. F. Song, C. Flindt, S. Rachel, I. Klich, and K. Le Hur, Entanglement entropy from charge statistics: Exact relations for noninteracting many-body systems, Phys. Rev. B 83, 161408 (2011).
  100. H. F. Song, S. Rachel, C. Flindt, I. Klich, N. Laflorencie, and K. Le Hur, Bipartite fluctuations as a probe of many-body entanglement, Phys. Rev. B 85, 035409 (2012).
  101. K. H. Thomas and C. Flindt, Entanglement entropy in dynamic quantum-coherent conductors, Phys. Rev. B 91, 125406 (2015).
  102. I. S. Burmistrov, K. S. Tikhonov, and I. V. Gornyi, Entanglement entropy and particle number cumulants of disordered fermions, Ann. Phys. (N. Y). 383, 140 (2017).
  103. Strictly speaking, the tensor product of fermionic Fock spaces is not well defined. However, one can give meaning to such a tensor product by first mapping the fermionic Fock spaces to bosonic ones through a Jordan-Wigner transformation, then forming the tensor product of the bosonic spaces, and finally reintroducing fermionic operators in the product space [45].
  104. B. Buča and T. Prosen, A note on symmetry reductions of the Lindblad equation: Transport in constrained open spin chains, New J. Phys. 14, 073007 (2012).
  105. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
  106. A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, Cambridge, 2010), 2nd ed.
  107. A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, 2023), 2nd ed.
  108. L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh field theory for driven open quantum systems, Rep. Prog. Phys. 79, 096001 (2016).
  109. L. M. Sieberer, M. Buchhold, J. Marino, and S. Diehl, Universality in driven open quantum matter, Rev. Mod. Phys. 97, 025004 (2025).
  110. One might wonder whether the replica Keldysh action agrees in the limit R→1 with the Keldysh action corresponding to the master equation that is obtained by averaging the stochastic Schrödinger equation (3) over the increments dNα,l(t). For occupation measurements, this is not the case. In the derivation of the replica Keldysh action, we have used the rate of jumps in Eq. (10). This rate reduces to γ only for N=L/2. Moreover, this rate is calculated using the operator identity ∑l=1Lψ^l†ψ^l=N. In contrast, in the master equation, the rate of jumps is encoded implicitly in the anticommutator in Eq. (21), which yields additional terms in the Keldysh action.
  111. This is a purely technical issue and does not imply that the dynamics is noncausal.
  112. According to the terminology introduced in Ref. [108], invariance of the Keldysh action under phase rotations with opposite signs on the forward and backward branches is an example of a quantum symmetry. For equal signs, we speak of a classical symmetry. In the operator formalism, a classical symmetry corresponds to a weak symmetry, and the combination of classical and quantum symmetries is equivalent to a strong symmetry [104, 105]. A definition of quantum symmetries that agrees with strong symmetries is adopted in Ref. [109].
  113. S. Hikami, Three-loop ß-functions of non-linear σ models on symmetric spaces, Phys. Lett. B 98, 208 (1981).
  114. F. Wegner, Four-loop-order β-function of nonlinear σ-models in symmetric spaces, Nucl. Phys. B 316, 663 (1989).
  115. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  116. P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. Theory Exp. 2004, P06002 (2004).
  117. I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A Math. Theor. 42, 504003 (2009).
  118. M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Area laws in quantum systems: Mutual information and correlations, Phys. Rev. Lett. 100, 070502 (2008).
  119. P. Calabrese, J. Cardy, and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. Theory Exp. 2009, P11001 (2009).
  120. V. Marić and M. Fagotti, Universality in the tripartite information after global quenches, Phys. Rev. B 108, L161116 (2023).
  121. V. Marić and M. Fagotti, Universality in the tripartite information after global quenches: (Generalised) quantum XY models, J. High Energy Phys. 2023, 140 (2023).
  122. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  123. A. Altland, M. Fleischhauer, and S. Diehl, Symmetry classes of open fermionic quantum matter, Phys. Rev. X 11, 021037 (2021).
  124. S. Lieu, M. McGinley, and N. R. Cooper, Tenfold way for quadratic Lindbladians, Phys. Rev. Lett. 124, 040401 (2020).
  125. L. Sá, P. Ribeiro, and T. Prosen, Symmetry classification of many-body Lindbladians: Tenfold way and beyond, Phys. Rev. X 13, 031019 (2023).
  126. K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, and S. Ryu, Symmetry of open quantum systems: Classification of dissipative quantum chaos, PRX Quantum 4, 030328 (2023).
  127. E. Starchl, M. H. Fischer, and L. M. Sieberer, Generalized Zeno effect and entanglement dynamics induced by fermion counting, [dataset] Universitt Innsbruck, 2025.
  128. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).

Outline

Information

PRX Quantum 6, 030302– Published 2 July, 2025
  • Received 19 June 2024
  • Revised 12 February 2025
  • Accepted 6 June 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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation