Controlling measurement-induced phase transitions with tunable detector coupling

Ritu Nehra; Alessandro Romito; Dganit Meidan

doi:10.22331/q-2025-04-08-1697

Controlling measurement-induced phase transitions with tunable detector coupling

Ritu Nehra¹, Alessandro Romito², and Dganit Meidan^1,3

¹Department of Physics, Ben Gurion University of the Negev, Israel
²Department of Physics, Lancaster University, United Kingdom
³Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France.

Published:	2025-04-08, volume 9, page 1697
Editor:	Angelo Carollo
Eprint:	arXiv:2404.07918v4
Doi:	https://doi.org/10.22331/q-2025-04-08-1697
Citation:	Quantum 9, 1697 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We study the evolution of a quantum many-body system driven by two competing measurements, which induces a topological entanglement transition between two distinct area law phases. We employ a positive operator-valued measurement with variable coupling between the system and detector within free fermion dynamics. This approach allows us to continuously track the universal properties of the transition between projective and continuous monitoring. Our findings suggest that the percolation universality of the transition in the projective limit is unstable when the system-detector coupling is reduced.

Featured image: (a) Space-time schematic of two competing measurements $\hat{M}^D$, $\hat{M}^K$ on the Majorana (free fermionic) chain. The two partitions of chains A and B contribute to calculating topological entanglement entropy. (b) The probability distribution of the detector outcome $x$ is modified by the coupling to the system. The modified probability distribution $ P(x)$ depends on the coupling strength ($\sqrt{\gamma dt}$).

Popular summary

The role of quantum measurements in shaping a system’s behavior has long been a subject of interest. This study examines how the coupling to the environment, which determines the amount of information an observer extracts, affects this behavior. We explore this in a system where the dynamics are driven by two competing quantum measurements, leading to a phase transition in the system’s entanglement properties.
At one extreme—strong, projective measurements—the system behaves like a classical percolation model, where entanglement is broken into isolated clusters that percolate at the transition. At the other extreme, weak and continuous monitoring—entanglement spreads more freely, as the observer causes minimal disturbance to the system’s state while also gathering very little information on average.
A key finding is that the mathematical characteristics of this transition evolve as measurement strength is tuned. In particular, the "correlation length exponent," which describes how correlations scale near the transition, steadily increases as the system moves away from the projective limit. This suggests that the classical percolation picture, valid under strong measurements, fails to describe the system as monitoring becomes weaker.

► BibTeX data

@article{Nehra2025controlling,
  doi = {10.22331/q-2025-04-08-1697},
  url = {https://doi.org/10.22331/q-2025-04-08-1697},
  title = {Controlling measurement-induced phase transitions with tunable detector coupling},
  author = {Nehra, Ritu and Romito, Alessandro and Meidan, Dganit},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1697},
  month = apr,
  year = {2025}
}

► References

[1] Kurt Jacobs. ``Quantum Measurement Theory and its Applications''. Cambridge University Press. Cambridge (2014).
https://doi.org/10.1017/CBO9781139179027

[2] Bei Zeng, Xie Chen, Duan-Lu Zhou, and Xiao-Gang Wen. ``Quantum information meets quantum matter''. Quantum Science and Technology. Springer. New York, NY (2019).
https://doi.org/10.1007/978-1-4939-9084-9

[3] J. Eisert, M. Cramer, and M. B. Plenio. ``Colloquium : Area laws for the entanglement entropy''. Rev. Mod. Phys. 82, 277–306 (2010).
https://doi.org/10.1103/RevModPhys.82.277

[4] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. ``Quantum entanglement growth under random unitary dynamics''. Phys. Rev. X 7, 031016 (2017).
https://doi.org/10.1103/PhysRevX.7.031016

[5] Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. ``Quantum Zeno effect and the many-body entanglement transition''. Phys. Rev. B 98, 205136 (2018).
https://doi.org/10.1103/PhysRevB.98.205136

[6] Amos Chan, Rahul M. Nandkishore, Michael Pretko, and Graeme Smith. ``Unitary-projective entanglement dynamics''. Phys. Rev. B 99, 224307 (2019).
https://doi.org/10.1103/PhysRevB.99.224307

[7] Brian Skinner, Jonathan Ruhman, and Adam Nahum. ``Measurement-Induced Phase Transitions in the Dynamics of Entanglement''. Phys. Rev. X 9, 031009 (2019).
https://doi.org/10.1103/PhysRevX.9.031009

[8] M. Szyniszewski, A. Romito, and H. Schomerus. ``Entanglement transition from variable-strength weak measurements''. Phys. Rev. B 100, 064204 (2019).
https://doi.org/10.1103/PhysRevB.100.064204

[9] Soonwon Choi, Yimu Bao, Xiao-Liang Qi, and Ehud Altman. ``Quantum error correction in scrambling dynamics and measurement-induced phase transition''. Phys. Rev. Lett. 125, 030505 (2020).
https://doi.org/10.1103/PhysRevLett.125.030505

[10] Yimu Bao, Soonwon Choi, and Ehud Altman. ``Theory of the phase transition in random unitary circuits with measurements''. Phys. Rev. B 101, 104301 (2020).
https://doi.org/10.1103/PhysRevB.101.104301

[11] Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. ``Random Quantum Circuits''. Annual Review of Condensed Matter Physics 14, 335–379 (2023).
https://doi.org/10.1146/annurev-conmatphys-031720-030658

[12] Crystal Noel, Pradeep Niroula, Daiwei Zhu, Andrew Risinger, Laird Egan, Debopriyo Biswas, Marko Cetina, Alexey V Gorshkov, Michael J Gullans, David A Huse, et al. ``Measurement-induced quantum phases realized in a trapped-ion quantum computer''. Nature Physics 18, 760–764 (2022).
https://doi.org/10.1038/s41567-022-01619-7

[13] Google Quantum AI and Collaborators. ``Measurement-induced entanglement and teleportation on a noisy quantum processor''. Nature 622, 481–486 (2023).
https://doi.org/10.1038/s41586-023-06505-7

[14] Jin Ming Koh, Shi-Ning Sun, Mario Motta, and Austin J Minnich. ``Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout''. Nature Physics 19, 1314–1319 (2023).
https://doi.org/10.1038/s41567-023-02076-6

[15] Nicolai Lang and Hans Peter Büchler. ``Entanglement transition in the projective transverse field Ising model''. Phys. Rev. B 102, 094204 (2020).
https://doi.org/10.1103/PhysRevB.102.094204

[16] Sthitadhi Roy, J. T. Chalker, I. V. Gornyi, and Yuval Gefen. ``Measurement-induced steering of quantum systems''. Phys. Rev. Research 2, 033347 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033347

[17] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli. ``Measurement-induced topological entanglement transitions in symmetric random quantum circuits''. Nat. Phys. 17, 342–347 (2021).
https://doi.org/10.1038/s41567-020-01112-z

[18] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli. ``Topological Order and Criticality in $\mathrm{(2+1)D}$ Monitored Random Quantum Circuits''. Phys. Rev. Lett. 127, 235701 (2021).
https://doi.org/10.1103/PhysRevLett.127.235701

[19] Matteo Ippoliti, Michael J. Gullans, Sarang Gopalakrishnan, David A. Huse, and Vedika Khemani. ``Entanglement phase transitions in measurement-only dynamics''. Phys. Rev. X 11, 011030 (2021).
https://doi.org/10.1103/PhysRevX.11.011030

[20] Shengqi Sang and Timothy H. Hsieh. ``Measurement-protected quantum phases''. Phys. Rev. Research 3, 023200 (2021).
https://doi.org/10.1103/PhysRevResearch.3.023200

[21] Graham Kells, Dganit Meidan, and Alessandro Romito. ``Topological transitions in weakly monitored free fermions''. SciPost Phys. 14, 031 (2023).
https://doi.org/10.21468/SciPostPhys.14.3.031

[22] Kai Klocke, Daniel Simm, Guo-Yi Zhu, Simon Trebst, and Michael Buchhold. ``Entanglement dynamics in monitored kitaev circuits: loop models, symmetry classification, and quantum lifshitz scaling'' (2024). arXiv:2409.02171.
arXiv:2409.02171

[23] A.Yu. Kitaev. ``Fault-tolerant quantum computation by anyons''. Annals of Physics 303, 2–30 (2003).
https://doi.org/10.1016/S0003-4916(02)00018-0

[24] Alexei Kitaev. ``Anyons in an exactly solved model and beyond''. Annals of Physics 321, 2–111 (2006).
https://doi.org/10.1016/j.aop.2005.10.005

[25] Kai Klocke and Michael Buchhold. ``Majorana loop models for measurement-only quantum circuits''. Phys. Rev. X 13, 041028 (2023).
https://doi.org/10.1103/PhysRevX.13.041028

[26] Michele Fava, Lorenzo Piroli, Tobias Swann, Denis Bernard, and Adam Nahum. ``Nonlinear sigma models for monitored dynamics of free fermions''. Phys. Rev. X 13, 041045 (2023).
https://doi.org/10.1103/PhysRevX.13.041045

[27] Igor Poboiko, Paul Pöpperl, Igor V. Gornyi, and Alexander D. Mirlin. ``Theory of free fermions under random projective measurements''. Phys. Rev. X 13, 041046 (2023).
https://doi.org/10.1103/PhysRevX.13.041046

[28] Zack Weinstein, Rohith Sajith, Ehud Altman, and Samuel J. Garratt. ``Nonlocality and entanglement in measured critical quantum ising chains''. Phys. Rev. B 107, 245132 (2023).
https://doi.org/10.1103/PhysRevB.107.245132

[29] Igor Poboiko, Igor V. Gornyi, and Alexander D. Mirlin. ``Measurement-induced phase transition for free fermions above one dimension''. Phys. Rev. Lett. 132, 110403 (2024).
https://doi.org/10.1103/PhysRevLett.132.110403

[30] Michele Coppola, Emanuele Tirrito, Dragi Karevski, and Mario Collura. ``Growth of entanglement entropy under local projective measurements''. Phys. Rev. B 105, 094303 (2022).
https://doi.org/10.1103/PhysRevB.105.094303

[31] Chao-Ming Jian, Hassan Shapourian, Bela Bauer, and Andreas W. W. Ludwig. ``Measurement-induced entanglement transitions in quantum circuits of non-interacting fermions: Born-rule versus forced measurements'' (2023). arXiv:2302.09094.
arXiv:2302.09094

[32] Luca Lumia, Emanuele Tirrito, Rosario Fazio, and Mario Collura. ``Measurement-induced transitions beyond gaussianity: A single particle description''. Phys. Rev. Res. 6, 023176 (2024).
https://doi.org/10.1103/PhysRevResearch.6.023176

[33] Elias Starchl, Mark H. Fischer, and Lukas M. Sieberer. ``Generalized zeno effect and entanglement dynamics induced by fermion counting'' (2025). arXiv:2406.07673.
arXiv:2406.07673

[34] Xiangyu Cao, Antoine Tilloy, and Andrea De Luca. ``Entanglement in a Fermion chain under continuous monitoring''. SciPost Physics 7, 024 (2019).
https://doi.org/10.21468/SciPostPhys.7.2.024

[35] Marcin Szyniszewski, Alessandro Romito, and Henning Schomerus. ``Universality of entanglement transitions from stroboscopic to continuous measurements''. Phys. Rev. Lett. 125, 210602 (2020).
https://doi.org/10.1103/PhysRevLett.125.210602

[36] 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).
https://doi.org/10.1103/PhysRevLett.126.170602

[37] Xhek Turkeshi, Alberto Biella, Rosario Fazio, Marcello Dalmonte, and Marco Schiró. ``Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks''. Phys. Rev. B 103, 224210 (2021).
https://doi.org/10.1103/PhysRevB.103.224210

[38] Alberto Biella and Marco Schiró. ``Many-Body Quantum Zeno Effect and Measurement-Induced Subradiance Transition''. Quantum 5, 528 (2021).
https://doi.org/10.22331/q-2021-08-19-528

[39] Youenn Le Gal, Xhek Turkeshi, and Marco Schirò. ``Entanglement dynamics in monitored systems and the role of quantum jumps''. PRX Quantum 5, 030329 (2024).
https://doi.org/10.1103/PRXQuantum.5.030329

[40] Bo Xing, Xhek Turkeshi, Marco Schiró, Rosario Fazio, and Dario Poletti. ``Interactions and integrability in weakly monitored hamiltonian systems''. Phys. Rev. B 109, L060302 (2024).
https://doi.org/10.1103/PhysRevB.109.L060302

[41] K. Chahine and M. Buchhold. ``Entanglement phases, localization, and multifractality of monitored free fermions in two dimensions''. Phys. Rev. B 110, 054313 (2024).
https://doi.org/10.1103/PhysRevB.110.054313

[42] Rafael D. Soares, Youenn Le Gal, and Marco Schirò. ``Entanglement transition due to particle losses in a monitored fermionic chain''. Phys. Rev. B 111, 064313 (2025).
https://doi.org/10.1103/PhysRevB.111.064313

[43] Tony Jin and David G. Martin. ``Measurement-induced phase transition in a single-body tight-binding model''. Phys. Rev. B 110, L060202 (2024).
https://doi.org/10.1103/PhysRevB.110.L060202

[44] Michele Fava, Lorenzo Piroli, Denis Bernard, and Adam Nahum. ``Monitored fermions with conserved $u(1)$ charge''. Phys. Rev. Res. 6, 043246 (2024).
https://doi.org/10.1103/PhysRevResearch.6.043246

[45] Marcin Szyniszewski. ``Unscrambling of single-particle wave functions in systems localized through disorder and monitoring''. Phys. Rev. B 110, 024303 (2024).
https://doi.org/10.1103/PhysRevB.110.024303

[46] Guillaume Cecile, Hugo Lóio, and Jacopo De Nardis. ``Measurement-induced phase transitions by matrix product states scaling''. Phys. Rev. Res. 6, 033220 (2024).
https://doi.org/10.1103/PhysRevResearch.6.033220

[47] 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).
https://doi.org/10.1103/PhysRevX.11.041004

[48] Federico Carollo and Vincenzo Alba. ``Entangled multiplets and spreading of quantum correlations in a continuously monitored tight-binding chain''. Phys. Rev. B 106, L220304 (2022).
https://doi.org/10.1103/PhysRevB.106.L220304

[49] Marcin Szyniszewski, Oliver Lunt, and Arijeet Pal. ``Disordered monitored free fermions''. Phys. Rev. B 108, 165126 (2023).
https://doi.org/10.1103/PhysRevB.108.165126

[50] Chun Y. Leung, Dganit Meidan, and Alessandro Romito. ``Theory of free fermions dynamics under partial post-selected monitoring'' (2023). arXiv:2312.14022.
arXiv:2312.14022

[51] Howard E. Brandt. ``Positive operator valued measure in quantum information processing''. American Journal of Physics 67, 434–439 (1999).
https://doi.org/10.1119/1.19280

[52] Howard M. Wiseman and Gerard J. Milburn. ``Quantum Measurement and Control''. Cambridge University Press. Cambridge (2009).
https://doi.org/10.1017/CBO9780511813948

[53] Alberto Barchielli and Matteo Gregoratti. ``Quantum Trajectories and Measurements in Continuous Time: The Diffusive Case''. Volume 782 of Lecture Notes in Physics. Springer. Berlin, Heidelberg (2009).
https://doi.org/10.1007/978-3-642-01298-3

[54] Joseph Merritt and Lukasz Fidkowski. ``Entanglement transitions with free fermions''. Phys. Rev. B 107, 064303 (2023).
https://doi.org/10.1103/PhysRevB.107.064303

[55] Alexei Kitaev and John Preskill. ``Topological Entanglement Entropy''. Phys. Rev. Lett. 96, 110404 (2006).
https://doi.org/10.1103/PhysRevLett.96.110404

[56] Michael Levin and Xiao-Gang Wen. ``Detecting topological order in a ground state wave function''. Phys. Rev. Lett. 96, 110405 (2006).
https://doi.org/10.1103/PhysRevLett.96.110405

[57] Bei Zeng and D. L. Zhou. ``Topological and error-correcting properties for symmetry-protected topological order''. EPL 113, 56001 (2016).
https://doi.org/10.1209/0295-5075/113/56001

[58] P. Fromholz, G. Magnifico, V. Vitale, T. Mendes-Santos, and M. Dalmonte. ``Entanglement topological invariants for one-dimensional topological superconductors''. Phys. Rev. B 101, 085136 (2020).
https://doi.org/10.1103/PhysRevB.101.085136

[59] Tommaso Micallo, Vittorio Vitale, Marcello Dalmonte, and Pierre Fromholz. ``Topological entanglement properties of disconnected partitions in the Su-Schrieffer-Heeger model''. SciPost Phys. Core 3, 012 (2020).
https://doi.org/10.21468/SciPostPhysCore.3.2.012

[60] Saikat Mondal, Souvik Bandyopadhyay, Sourav Bhattacharjee, and Amit Dutta. ``Detecting topological phase transitions through entanglement between disconnected partitions in a Kitaev chain with long-range interactions''. Phys. Rev. B 105, 085106 (2022).
https://doi.org/10.1103/PhysRevB.105.085106

[61] Ingo Peschel. ``Calculation of reduced density matrices from correlation functions''. J. Phys. A: Math. Gen. 36, L205 (2003).
https://doi.org/10.1088/0305-4470/36/14/101

[62] J.P. Keating and F. Mezzadri. ``Random Matrix Theory and Entanglement in Quantum Spin Chains''. Commun. Math. Phys. 252, 543–579 (2004).
https://doi.org/10.1007/s00220-004-1188-2

[63] Ingo Peschel and Viktor Eisler. ``Reduced density matrices and entanglement entropy in free lattice models''. J. Phys. A: Math. Theor. 42, 504003 (2009).
https://doi.org/10.1088/1751-8113/42/50/504003

[64] Ritu Nehra, Devendra Singh Bhakuni, Ajith Ramachandran, and Auditya Sharma. ``Flat bands and entanglement in the Kitaev ladder''. Phys. Rev. Res. 2, 013175 (2020).
https://doi.org/10.1103/PhysRevResearch.2.013175

[65] M. F. Sykes and J. W. Essam. ``Exact Critical Percolation Probabilities for Site and Bond Problems in Two Dimensions''. Journal of Mathematical Physics 5, 1117–1127 (1964).
https://doi.org/10.1063/1.1704215

[66] Youjin Deng and Xianqing Yang. ``Finite-size scaling of energylike quantities in percolation''. Phys. Rev. E 73, 066116 (2006).
https://doi.org/10.1103/PhysRevE.73.066116

[67] Gergő Roósz, Róbert Juhász, and Ferenc Iglói. ``Nonequilibrium dynamics of the Ising chain in a fluctuating transverse field''. Phys. Rev. B 93, 134305 (2016).
https://doi.org/10.1103/PhysRevB.93.134305

[68] Dietrich Stauffer and Ammon Aharony. ``Introduction to percolation theory''. CRC press. (2018).
https://doi.org/10.1201/9781315274386

[69] Kemal Aziz, Ahana Chakraborty, and J. H. Pixley. ``Critical properties of weak measurement induced phase transitions in random quantum circuits''. Phys. Rev. B 110, 064301 (2024).
https://doi.org/10.1103/PhysRevB.110.064301

[70] Elmer V. H. Doggen, Igor V. Gornyi, and Alexander D. Mirlin. ``Ancilla quantum measurements on interacting chains: Sensitivity of entanglement dynamics to the type and concentration of detectors''. Phys. Rev. B 109, 224203 (2024).
https://doi.org/10.1103/PhysRevB.109.224203

[71] Michael M. Wolf, Frank Verstraete, Matthew B. Hastings, and J. Ignacio Cirac. ``Area Laws in Quantum Systems: Mutual Information and Correlations''. Phys. Rev. Lett. 100, 070502 (2008).
https://doi.org/10.1103/PhysRevLett.100.070502

Cited by

[1] Surajit Bera, Igor V. Gornyi, Sumilan Banerjee, and Yuval Gefen, "Generation of volume-law entanglement by local-measurement-only quantum dynamics", Physical Review B 113 14, 144309 (2026).

[2] Hisanori Oshima, Ken Mochizuki, Ryusuke Hamazaki, and Yohei Fuji, "Topology and Spectrum in Measurement-Induced Phase Transitions", Physical Review Letters 134 24, 240401 (2025).

[3] Giulia Piccitto, Giuliano Chiriacò, Davide Rossini, and Angelo Russomanno, "Entanglement behavior and localization properties in monitored fermion systems", Physical Review B 112 17, 174311 (2025).

[4] Giuseppe Di Giulio, Xhek Turkeshi, and Sara Murciano, "Measurement-Induced Symmetry Restoration and Quantum Mpemba Effect", Entropy 27 4, 407 (2025).

[5] Igor Poboiko, Marcin Szyniszewski, Christopher J. Turner, Igor V. Gornyi, Alexander D. Mirlin, and Arijeet Pal, "Measurement-Induced Lévy Flights of Quantum Information", Physical Review Letters 135 17, 170403 (2025).

[6] Ryusuke Hamazaki, Ken Mochizuki, Hisanori Oshima, and Yohei Fuji, "An Introduction to Monitored Quantum Systems and Quantum Trajectories: Spectrum, Typicality, and Phases", Progress of Theoretical and Experimental Physics 2026 5, 052A01 (2026).

[7] Cristiano Muzzi, Mikheil Tsitsishvili, and Giuliano Chiriacò, "Entanglement enhancement induced by noise in inhomogeneously monitored systems", Physical Review B 111 1, 014312 (2025).

[8] Kemal Aziz, Ahana Chakraborty, and J. H. Pixley, "Critical properties of weak measurement induced phase transitions in random quantum circuits", Physical Review B 110 6, 064301 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 06:05:19) and SAO/NASA ADS (last updated successfully 2026-08-11 14:34:46). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-08-12 06:05:19: Cannot retrieve data from ADS due to rate limitations.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.