{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T19:26:56Z","timestamp":1779305216650,"version":"3.51.4"},"reference-count":66,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T00:00:00Z","timestamp":1750118400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Symmetry is a powerful tool for understanding phases of matter in equilibrium. Quantum circuits with measurements have recently emerged as a platform for novel states of matter intrinsically out of equilibrium. Can symmetry be used as an organizing principle for these novel states, their phases and phase transitions? In this work, we give an affirmative answer to this question in a simple adaptive monitored circuit, which hosts an ordering transition in addition to a separate entanglement transition, upon tuning a single parameter. Starting from a symmetry-breaking initial state, depending on the tuning parameter, the steady state could (i) remain symmetry-broken, (ii) exhibit the average symmetry in the ensemble of trajectories, or (iii) exhibit the exact symmetry for each trajectory. The ordering transition is mapped to the transition in a classical majority vote model, described by the Ising universality class, while the entanglement transition lies in the percolation class. Numerical simulations are further presented to support the analytical understandings.<\/jats:p>","DOI":"10.22331\/q-2025-06-17-1771","type":"journal-article","created":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T10:53:31Z","timestamp":1750157611000},"page":"1771","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Exact, Average, and Broken Symmetries in a Simple Adaptive Monitored Circuit"],"prefix":"10.22331","volume":"9","author":[{"given":"Zhi","family":"Li","sequence":"first","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhu-Xi","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge MA-02138, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,6,17]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Matthew P.A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. ``Random quantum circuits''. Annual Review of Condensed Matter Physics 14, 335\u2013379 (2023).","DOI":"10.1146\/annurev-conmatphys-031720-030658"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Brian Skinner, Jonathan Ruhman, and Adam Nahum. ``Measurement-induced phase transitions in the dynamics of entanglement''. Phys. Rev. X 9, 031009 (2019).","DOI":"10.1103\/PhysRevX.9.031009"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. ``Quantum zeno effect and the many-body entanglement transition''. Phys. Rev. B 98, 205136 (2018).","DOI":"10.1103\/PhysRevB.98.205136"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Amos Chan, Rahul M Nandkishore, Michael Pretko, and Graeme Smith. ``Unitary-projective entanglement dynamics''. Physical Review B 99, 224307 (2019).","DOI":"10.1103\/PhysRevB.99.224307"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Michael J Gullans and David A Huse. ``Dynamical purification phase transition induced by quantum measurements''. Physical Review X 10, 041020 (2020).","DOI":"10.1103\/PhysRevX.10.041020"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Yaodong Li, Xiao Chen, and Matthew PA Fisher. ``Measurement-driven entanglement transition in hybrid quantum circuits''. Physical Review B 100, 134306 (2019).","DOI":"10.1103\/PhysRevB.100.134306"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Aidan Zabalo, Michael J Gullans, Justin H Wilson, Sarang Gopalakrishnan, David A Huse, and JH Pixley. ``Critical properties of the measurement-induced transition in random quantum circuits''. Physical Review B 101, 060301 (2020).","DOI":"10.1103\/PhysRevB.101.060301"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Chao-Ming Jian, Yi-Zhuang You, Romain Vasseur, and Andreas W. W. Ludwig. ``Measurement-induced criticality in random quantum circuits''. Phys. Rev. B 101, 104302 (2020).","DOI":"10.1103\/PhysRevB.101.104302"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Yimu Bao, Soonwon Choi, and Ehud Altman. ``Theory of the phase transition in random unitary circuits with measurements''. Phys. Rev. B 101, 104301 (2020).","DOI":"10.1103\/PhysRevB.101.104301"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Michael J. Gullans and David A. Huse. ``Scalable probes of measurement-induced criticality''. Phys. Rev. Lett. 125, 070606 (2020).","DOI":"10.1103\/PhysRevLett.125.070606"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Beni Yoshida. ``Decoding the entanglement structure of monitored quantum circuits''. arXiv:2109.08691 (2021).","DOI":"10.48550\/arXiv.2109.08691"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Shengqi Sang, Zhi Li, Timothy H. Hsieh, and Beni Yoshida. ``Ultrafast entanglement dynamics in monitored quantum circuits''. PRX Quantum 4, 040332 (2023).","DOI":"10.1103\/PRXQuantum.4.040332"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Adam Nahum and Brian Skinner. ``Entanglement and dynamics of diffusion-annihilation processes with majorana defects''. Phys. Rev. Res. 2, 023288 (2020).","DOI":"10.1103\/PhysRevResearch.2.023288"},{"key":"13","doi-asserted-by":"publisher","unstructured":"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).","DOI":"10.1103\/PhysRevX.11.011030"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli. ``Measurement-induced topological entanglement transitions in symmetric random quantum circuits''. Nature Physics 17, 342\u2013347 (2021).","DOI":"10.1038\/s41567-020-01112-z"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Shengqi Sang and Timothy H. Hsieh. ``Measurement-protected quantum phases''. Phys. Rev. Res. 3, 023200 (2021).","DOI":"10.1103\/PhysRevResearch.3.023200"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Nicolai Lang and Hans Peter B\u00fcchler. ``Entanglement transition in the projective transverse field Ising model''. Phys. Rev. B 102, 094204 (2020).","DOI":"10.1103\/PhysRevB.102.094204"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Ali Lavasani, Zhu-Xi Luo, and Sagar Vijay. ``Monitored quantum dynamics and the kitaev spin liquid''. Phys. Rev. B 108, 115135 (2023).","DOI":"10.1103\/PhysRevB.108.115135"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Adithya Sriram, Tibor Rakovszky, Vedika Khemani, and Matteo Ippoliti. ``Topology, criticality, and dynamically generated qubits in a stochastic measurement-only kitaev model''. Phys. Rev. B 108, 094304 (2023).","DOI":"10.1103\/PhysRevB.108.094304"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Guo-Yi Zhu, Nathanan Tantivasadakarn, and Simon Trebst. ``Structured volume-law entanglement in an interacting, monitored majorana spin liquid''. Phys. Rev. Res. 6, L042063 (2024).","DOI":"10.1103\/PhysRevResearch.6.L042063"},{"key":"20","doi-asserted-by":"publisher","unstructured":"P. W. Anderson. ``More is different''. Science 177, 393\u2013396 (1972).","DOI":"10.1126\/science.177.4047.393"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Adam Nahum, Sthitadhi Roy, Brian Skinner, and Jonathan Ruhman. ``Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in landau-ginsburg theory''. PRX Quantum 2, 010352 (2021).","DOI":"10.1103\/PRXQuantum.2.010352"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Yiqiu Han and Xiao Chen. ``Measurement-induced criticality in $\\mathbb{Z}_{2}$-symmetric quantum automaton circuits''. Phys. Rev. B 105, 064306 (2022).","DOI":"10.1103\/PhysRevB.105.064306"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Lorenzo Piroli, Yaodong Li, Romain Vasseur, and Adam Nahum. ``Triviality of quantum trajectories close to a directed percolation transition''. Phys. Rev. B 107, 224303 (2023).","DOI":"10.1103\/PhysRevB.107.224303"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Vikram Ravindranath, Yiqiu Han, Zhi-Cheng Yang, and Xiao Chen. ``Entanglement steering in adaptive circuits with feedback''. Phys. Rev. B 108, L041103 (2023).","DOI":"10.1103\/PhysRevB.108.L041103"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Nicholas O'Dea, Alan Morningstar, Sarang Gopalakrishnan, and Vedika Khemani. ``Entanglement and absorbing-state transitions in interactive quantum dynamics''. Phys. Rev. B 109, L020304 (2024).","DOI":"10.1103\/PhysRevB.109.L020304"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Jason Iaconis, Andrew Lucas, and Xiao Chen. ``Measurement-induced phase transitions in quantum automaton circuits''. Phys. Rev. B 102, 224311 (2020).","DOI":"10.1103\/PhysRevB.102.224311"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Utkarsh Agrawal, Aidan Zabalo, Kun Chen, Justin H. Wilson, Andrew C. Potter, J. H. Pixley, Sarang Gopalakrishnan, and Romain Vasseur. ``Entanglement and charge-sharpening transitions in u(1) symmetric monitored quantum circuits''. Phys. Rev. X 12, 041002 (2022).","DOI":"10.1103\/PhysRevX.12.041002"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Fergus Barratt, Utkarsh Agrawal, Sarang Gopalakrishnan, David A. Huse, Romain Vasseur, and Andrew C. Potter. ``Field theory of charge sharpening in symmetric monitored quantum circuits''. Phys. Rev. Lett. 129, 120604 (2022).","DOI":"10.1103\/PhysRevLett.129.120604"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Fergus Barratt, Utkarsh Agrawal, Andrew C. Potter, Sarang Gopalakrishnan, and Romain Vasseur. ``Transitions in the learnability of global charges from local measurements''. Phys. Rev. Lett. 129, 200602 (2022).","DOI":"10.1103\/PhysRevLett.129.200602"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Hisanori Oshima and Yohei Fuji. ``Charge fluctuation and charge-resolved entanglement in a monitored quantum circuit with $u(1)$ symmetry''. Phys. Rev. B 107, 014308 (2023).","DOI":"10.1103\/PhysRevB.107.014308"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Ali G. Moghaddam, Kim P\u00f6yh\u00f6nen, and Teemu Ojanen. ``Exponential shortcut to measurement-induced entanglement phase transitions''. Phys. Rev. Lett. 131, 020401 (2023).","DOI":"10.1103\/PhysRevLett.131.020401"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Yimu Bao, Soonwon Choi, and Ehud Altman. ``Symmetry enriched phases of quantum circuits''. Annals of Physics 435, 168618 (2021).","DOI":"10.1016\/j.aop.2021.168618"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Felix Roser, Hans Peter B\u00fcchler, and Nicolai Lang. ``Decoding the projective transverse field ising model''. Phys. Rev. B 107, 214201 (2023).","DOI":"10.1103\/PhysRevB.107.214201"},{"key":"34","doi-asserted-by":"publisher","unstructured":"Yaodong Li and Matthew P. A. Fisher. ``Decodable hybrid dynamics of open quantum systems with ${\\mathbb{Z}}_{2}$ symmetry''. Phys. Rev. B 108, 214302 (2023).","DOI":"10.1103\/PhysRevB.108.214302"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Amir-Reza Negari, Subhayan Sahu, and Timothy H. Hsieh. ``Measurement-induced phase transitions in the toric code''. Phys. Rev. B 109, 125148 (2024).","DOI":"10.1103\/PhysRevB.109.125148"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Ruochen Ma and Chong Wang. ``Average Symmetry-Protected Topological Phases''. Phys. Rev. X 13, 031016 (2023). arXiv:2209.02723.","DOI":"10.1103\/PhysRevX.13.031016"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Jian-Hao Zhang, Yang Qi, and Zhen Bi. ``Strange correlation function for average symmetry-protected topological phases''. arXiv:2210.17485 (2022).","DOI":"10.48550\/arXiv.2210.17485"},{"key":"38","doi-asserted-by":"publisher","unstructured":"Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng, and Chong Wang. ``Topological phases with average symmetries: The decohered, the disordered, and the intrinsic''. Phys. Rev. X 15, 021062 (2025).","DOI":"10.1103\/PhysRevX.15.021062"},{"key":"39","doi-asserted-by":"publisher","unstructured":"Jong Yeon Lee, Yi-Zhuang You, and Cenke Xu. ``Symmetry protected topological phases under decoherence''. Quantum 9, 1607 (2025).","DOI":"10.22331\/q-2025-01-23-1607"},{"key":"40","doi-asserted-by":"publisher","unstructured":"F. Grusdt. ``Topological order of mixed states in correlated quantum many-body systems''. Phys. Rev. B 95, 075106 (2017).","DOI":"10.1103\/PhysRevB.95.075106"},{"key":"41","doi-asserted-by":"publisher","unstructured":"Po-Shen Hsin, Zhu-Xi Luo, and Hao-Yu Sun. ``Anomalies of average symmetries: Entanglement and open quantum systems''. Journal of High Energy Physics 2024, 1\u201342 (2024).","DOI":"10.1007\/JHEP10(2024)134"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Yi-Neng Zhou, Xingyu Li, Hui Zhai, Chengshu Li, and Yingfei Gu. ``Reviving the lieb\u2013schultz\u2013mattis theorem in open quantum systems''. National Science Review 12, nwae287 (2025).","DOI":"10.1093\/nsr\/nwae287"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Yunlong Zang, Yingfei Gu, and Shenghan Jiang. ``Detecting quantum anomalies in open systems''. Phys. Rev. Lett. 133, 106503 (2024).","DOI":"10.1103\/PhysRevLett.133.106503"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Michael Buchhold, Thomas Mueller, and Sebastian Diehl. ``Revealing measurement-induced phase transitions by pre-selection''. arXiv:2208.10506 (2022).","DOI":"10.48550\/arXiv.2208.10506"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Charles H. Bennett and G. Grinstein. ``Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems''. Phys. Rev. Lett. 55, 657\u2013660 (1985).","DOI":"10.1103\/PhysRevLett.55.657"},{"key":"46","doi-asserted-by":"publisher","unstructured":"T\u00e2nia Tom\u00e9 and M\u00e1rio J De Oliveira. ``Stochastic dynamics and irreversibility''. Springer. (2015).","DOI":"10.1007\/978-3-319-11770-6"},{"key":"47","doi-asserted-by":"publisher","unstructured":"M. J. de Oliveira. ``Isotropic majority-vote model on a square lattice''. Journal of Statistical Physics 66, 273\u2013281 (1992).","DOI":"10.1007\/BF01060069"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Cristopher Moore. ``Majority-vote cellular automata, Ising dynamics, and p-completeness''. Journal of Statistical Physics 88, 795\u2013805 (1997).","DOI":"10.1023\/B:JOSS.0000015172.31951.7b"},{"key":"49","doi-asserted-by":"publisher","unstructured":"Max McGinley, Sthitadhi Roy, and S. A. Parameswaran. ``Absolutely stable spatiotemporal order in noisy quantum systems''. Phys. Rev. Lett. 129, 090404 (2022).","DOI":"10.1103\/PhysRevLett.129.090404"},{"key":"50","doi-asserted-by":"publisher","unstructured":"Daniel Gottesman. ``Stabilizer codes and quantum error correction''. California Institute of Technology. (1997).","DOI":"10.48550\/arXiv.quant-ph\/9705052"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Christian D. Lorenz and Robert M. Ziff. ``Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices''. Phys. Rev. E 57, 230\u2013236 (1998).","DOI":"10.1103\/PhysRevE.57.230"},{"key":"52","doi-asserted-by":"publisher","unstructured":"Junfeng Wang, Zongzheng Zhou, Wei Zhang, Timothy M. Garoni, and Youjin Deng. ``Bond and site percolation in three dimensions''. Phys. Rev. E 87, 052107 (2013).","DOI":"10.1103\/PhysRevE.87.052107"},{"key":"53","doi-asserted-by":"publisher","unstructured":"Xiao Xu, Junfeng Wang, Jian-Ping Lv, and Youjin Deng. ``Simultaneous analysis of three-dimensional percolation models''. Frontiers of Physics 9, 113\u2013119 (2014).","DOI":"10.1007\/s11467-013-0403-z"},{"key":"54","doi-asserted-by":"publisher","unstructured":"R. Peierls. ``On Ising's model of ferromagnetism''. Mathematical Proceedings of the Cambridge Philosophical Society 32, 477\u2013481 (1936).","DOI":"10.1017\/S0305004100019174"},{"key":"55","doi-asserted-by":"publisher","unstructured":"Robert B. Griffiths. ``Peierls proof of spontaneous magnetization in a two-dimensional Ising ferromagnet''. Phys. Rev. 136, A437\u2013A439 (1964).","DOI":"10.1103\/PhysRev.136.A437"},{"key":"56","unstructured":"Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. ``Statistical physics: Volume 5''. Volume 5. Elsevier. (2013)."},{"key":"57","doi-asserted-by":"publisher","unstructured":"Peter G\u00e1cs. ``Reliable computation with cellular automata''. Journal of Computer and System Sciences 32, 15\u201378 (1986).","DOI":"10.1016\/0022-0000(86)90002-4"},{"key":"58","doi-asserted-by":"publisher","unstructured":"Lawrence F Gray. ``A reader's guide to gacs's \u201cpositive rates\u201d paper''. Journal of Statistical Physics 103, 1\u201344 (2001).","DOI":"10.1023\/A:1004824203467"},{"key":"59","doi-asserted-by":"publisher","unstructured":"Jong Yeon Lee, Chao-Ming Jian, and Cenke Xu. ``Quantum criticality under decoherence or weak measurement''. PRX Quantum 4, 030317 (2023).","DOI":"10.1103\/PRXQuantum.4.030317"},{"key":"60","doi-asserted-by":"publisher","unstructured":"Leonardo A. Lessa, Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng, and Chong Wang. ``Strong-to-weak spontaneous symmetry breaking in mixed quantum states''. PRX Quantum 6, 010344 (2025).","DOI":"10.1103\/PRXQuantum.6.010344"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Carolyn Zhang, Yichen Xu, Jian-Hao Zhang, Cenke Xu, Zhen Bi, and Zhu-Xi Luo. ``Strong-to-weak spontaneous breaking of 1-form symmetry and intrinsically mixed topological order''. arXiv:2409.17530 (2024).","DOI":"10.48550\/arXiv.2409.17530"},{"key":"62","doi-asserted-by":"publisher","unstructured":"Zeyu Liu, Langxuan Chen, Yuke Zhang, Shuyan Zhou, and Pengfei Zhang. ``Diagnosing strong-to-weak symmetry breaking via wightman correlators''. arXiv.2410.09327 (2024).","DOI":"10.48550\/arXiv.2410.09327"},{"key":"63","doi-asserted-by":"publisher","unstructured":"Shayan Majidy, Utkarsh Agrawal, Sarang Gopalakrishnan, Andrew C. Potter, Romain Vasseur, and Nicole Yunger Halpern. ``Critical phase and spin sharpening in su(2)-symmetric monitored quantum circuits''. Phys. Rev. B 108, 054307 (2023).","DOI":"10.1103\/PhysRevB.108.054307"},{"key":"64","doi-asserted-by":"publisher","unstructured":"Leonardo A. Lessa, Meng Cheng, and Chong Wang. ``Mixed-state quantum anomaly and multipartite entanglement''. Phys. Rev. X 15, 011069 (2025).","DOI":"10.1103\/PhysRevX.15.011069"},{"key":"65","doi-asserted-by":"publisher","unstructured":"Xue-Jia Yu, Sheng Yang, Shuo Liu, Hai-Qing Lin, and Shao-Kai Jian. ``Gapless symmetry-protected topological states in measurement-only circuits''. arXiv:2501.03851 (2025).","DOI":"10.48550\/arXiv.2501.03851"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-06-17-1771\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T10:53:34Z","timestamp":1750157614000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-06-17-1771\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,17]]},"references-count":66,"URL":"https:\/\/doi.org\/10.22331\/q-2025-06-17-1771","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,17]]},"article-number":"1771"}}