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This research investigates the spatial structure of multipartite entanglement near measurement-induced phase transitions, revealing long-range genuine multiparty correlations that differ fundamentally from equilibrium critical points. By mapping these quantum phenomena to classical percolation models via non-unitary conformal field theory, the study derives exact entanglement exponents for both 1D and 2D systems. These findings establish a rigorous framework for analyzing entanglement hierarchies in general quantum circuit ensembles, advancing theoretical understanding of quantum criticality.
Spatial structure of multipartite entanglement at measurement induced phase transitions
Quantum 10, 2211 (2026). https://doi.org/10.22331/q-2026-09-18-2211 We study multiparty entanglement near measurement induced phase transitions (MIPTs), which arise in ensembles of local quantum circuits built with unitaries and measurements. In contrast to equilibrium quantum critical transitions, where entanglement is short-ranged, MIPTs possess long-range k-party genuine multiparty entanglement (GME) characterized by an infinite hierarchy of entanglement exponents for k$\geq$2. First, we represent the average spread of entanglement with "entanglement clusters", and use them to conjecture general exponent relations: 1) classical dominance, 2) monotonicity, 3) subadditivity. We then introduce measure-weighted graphs to construct such clusters in general circuits. Second, we obtain the exact entanglement exponents for a 1d MIPT in a measurement-only circuit that maps to percolation by exploiting non-unitary conformal field theory. The exponents, which we numerically verify, obey the inequalities. We also extend the construction to a 2d MIPT that maps to classical 3d percolation, and numerically find the first entanglement exponents. Our results provide a firm ground to understand the multiparty entanglement of MIPTs, and more general ensembles of quantum circuits.
We study multiparty entanglement near measurement induced phase transitions (MIPTs), which arise in ensembles of local quantum circuits built with unitaries and measurements. In contrast to equilibrium quantum critical transitions, where entanglement is short-ranged, MIPTs possess long-range k-party genuine multiparty entanglement (GME) characterized by an infinite hierarchy of entanglement exponents for k$\geq$2. First, we represent the average spread of entanglement with "entanglement clusters", and use them to conjecture general exponent relations: 1) classical dominance, 2) monotonicity, 3) subadditivity. We then introduce measure-weighted graphs to construct such clusters in general circuits. Second, we obtain the exact entanglement exponents for a 1d MIPT in a measurement-only circuit that maps to percolation by exploiting non-unitary conformal field theory. The exponents, which we numerically verify, obey the inequalities. We also extend the construction to a 2d MIPT that maps to classical 3d percolation, and numerically find the first entanglement exponents. Our results provide a firm ground to understand the multiparty entanglement of MIPTs, and more general ensembles of quantum circuits.
[1] Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett., 86: 5188–5191, May 2001. 10.1103/PhysRevLett.86.5188. URL https://link.aps.org/doi/10.1103/PhysRevLett.86.5188. https://doi.org/10.1103/PhysRevLett.86.5188
[2] Anders S. Sørensen and Klaus Mølmer. Entanglement and extreme spin squeezing. Phys. Rev. Lett., 86: 4431–4434, May 2001. 10.1103/PhysRevLett.86.4431. URL https://link.aps.org/doi/10.1103/PhysRevLett.86.4431. https://doi.org/10.1103/PhysRevLett.86.4431
[3] A. J. Scott. Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions. Phys. Rev. A, 69: 052330, May 2004. 10.1103/PhysRevA.69.052330. URL https://link.aps.org/doi/10.1103/PhysRevA.69.052330. https://doi.org/10.1103/PhysRevA.69.052330
[4] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van Den Nest. Measurement-based quantum computation. Nature Physics, 5 (1): 19–36, 2009. ISSN 17452481. 10.1038/nphys1157. https://doi.org/10.1038/nphys1157
[5] A. Osterloh, Luigi Amico, G. Falci, and Rosario Fazio. Scaling of entanglement close to a quantum phase transition. Nature, 416: 608, 2002. 10.1038/416608a. https://doi.org/10.1038/416608a
[6] Tobias J. Osborne and Michael A. Nielsen. Entanglement in a simple quantum phase transition. Phys. Rev. A, 66: 032110, 2002. 10.1103/PhysRevA.66.032110. https://doi.org/10.1103/PhysRevA.66.032110
[7] Younes Javanmard, Daniele Trapin, Soumya Bera, Jens H. Bardarson, and Markus Heyl. Sharp entanglement thresholds in the logarithmic negativity of disjoint blocks in the transverse-field Ising chain. New Journal of Physics, 20 (8): 083032, 8 2018. ISSN 13672630. 10.1088/1367-2630/aad9ba. https://doi.org/10.1088/1367-2630/aad9ba
[8] Gilles Parez and William Witczak-Krempa. Entanglement negativity between separated regions in quantum critical systems. Physical Review Research, 6 (2): 023125, 4 2024. ISSN 26431564. 10.1103/PhysRevResearch.6.023125. https://doi.org/10.1103/PhysRevResearch.6.023125
[9] Gilles Parez and William Witczak-Krempa. The fate of entanglement. SciPost Physics, 20 (1): 002, 1 2026. 10.21468/SciPostPhys.20.1.002. URL http://arxiv.org/abs/2402.06677. https://doi.org/10.21468/SciPostPhys.20.1.002 arXiv:2402.06677
[10] Shengqi Sang, Yaodong Li, Tianci Zhou, Xiao Chen, Timothy H. Hsieh, and Matthew P. A. Fisher. Entanglement Negativity at Measurement-Induced Criticality. PRX Quantum, 2 (3): 030313, 7 2021. 10.1103/PRXQuantum.2.030313. URL http://arxiv.org/abs/2012.00031 http://dx.doi.org/10.1103/PRXQuantum.2.030313. https://doi.org/10.1103/PRXQuantum.2.030313 arXiv:2012.00031
[11] Sebastien J Avakian, T. Pereg-Barnea, and William Witczak-Krempa. Long-range multipartite entanglement near measurement-induced transitions. Physical Review Research, 7 (2): 023135, 5 2025. https://doi.org/10.1103/PhysRevResearch.7.023135. URL http://arxiv.org/abs/2404.16095. https://doi.org/10.1103/PhysRevResearch.7.023135 arXiv:2404.16095
[12] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. Quantum entanglement growth under random unitary dynamics. Physical Review X, 7 (3): 031016, 7 2017. ISSN 21603308. 10.1103/PhysRevX.7.031016. https://doi.org/10.1103/PhysRevX.7.031016
[13] Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani. “Quantum supremacy” and the complexity of random circuit sampling. Leibniz International Proceedings in Informatics, LIPIcs, 124, 2019a. ISSN 18688969. 10.4230/LIPIcs.ITCS.2019.15. https://doi.org/10.4230/LIPIcs.ITCS.2019.15
[14] Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani. On the complexity and verification of quantum random circuit sampling. Nature Physics, 15 (2): 159–163, 2019b. ISSN 17452481. 10.1038/s41567-018-0318-2. URL https://doi.org/10.1038/s41567-018-0318-2. https://doi.org/10.1038/s41567-018-0318-2
[15] Andrew C. Potter and Romain Vasseur. Entanglement Dynamics in Hybrid Quantum Circuits, pages 211–249. Springer, Cham, 4 2022. ISBN 978-3-031-03998-0. 10.1007/978-3-031-03998-0_9. URL https://link.springer.com/10.1007/978-3-031-03998-0_9. https://doi.org/10.1007/978-3-031-03998-0_9
[16] Matthew P. A. Fisher, Vedika Khemani, Adam Nahum, and Sagar Vijay. Random Quantum Circuits. Annual Review of Condensed Matter Physics, 14: 335–379, 3 2023. 10.1146/annurev-conmatphys-031720-030658. https://doi.org/10.1146/annurev-conmatphys-031720-030658
[17] Fernando G.S.L. Brandão, Aram W. Harrow, and Michał Horodecki. Local Random Quantum Circuits are Approximate Polynomial-Designs. Communications in Mathematical Physics, 346 (2): 397–434, 2016. ISSN 14320916. 10.1007/s00220-016-2706-8. https://doi.org/10.1007/s00220-016-2706-8
[18] Sergio Boixo, Sergei V. Isakov, Vadim N. Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J. Bremner, John M. Martinis, and Hartmut Neven. Characterizing quantum supremacy in near-term devices. Nature Physics, 14 (6): 595–600, 2018. ISSN 17452481. 10.1038/s41567-018-0124-x. https://doi.org/10.1038/s41567-018-0124-x
[19] John C. Napp, Rolando L. La Placa, Alexander M. Dalzell, Fernando G.S.L. Brandão, and Aram W. Harrow. Efficient Classical Simulation of Random Shallow 2D Quantum Circuits. Physical Review X, 12 (2): 021021, 4 2022. ISSN 21603308. 10.1103/PhysRevX.12.021021. https://doi.org/10.1103/PhysRevX.12.021021
[20] Yaodong Li, Xiao Chen, and Matthew P A Fisher. Quantum Zeno effect and the many-body entanglement transition. Physical Review B, 98 (20): 205136, 11 2018. 10.1103/PhysRevB.98.205136. https://doi.org/10.1103/PhysRevB.98.205136
[21] Amos Chan, Rahul M. Nandkishore, Michael Pretko, and Graeme Smith. Unitary-projective entanglement dynamics. Phys. Rev. B, 99: 224307, Jun 2019. 10.1103/PhysRevB.99.224307. URL https://link.aps.org/doi/10.1103/PhysRevB.99.224307. https://doi.org/10.1103/PhysRevB.99.224307
[22] Yaodong Li, Xiao Chen, and Matthew P.A. Fisher. Measurement-driven entanglement transition in hybrid quantum circuits. Physical Review B, 100 (13): 134306, 10 2019. ISSN 24699969. 10.1103/PhysRevB.100.134306. https://doi.org/10.1103/PhysRevB.100.134306
[23] Brian Skinner, Jonathan Ruhman, and Adam Nahum. Measurement-Induced Phase Transitions in the Dynamics of Entanglement. Physical Review X, 9 (3), 2019. ISSN 21603308. 10.1103/PhysRevX.9.031009. https://doi.org/10.1103/PhysRevX.9.031009
[24] Romain Vasseur, Andrew C. Potter, Yi Zhuang You, and Andreas W.W. Ludwig. Entanglement transitions from holographic random tensor networks. Physical Review B, 100 (13): 134203, 10 2019. ISSN 24699969. 10.1103/PhysRevB.100.134203. https://doi.org/10.1103/PhysRevB.100.134203
[25] Aidan Zabalo, Michael J. Gullans, Justin H. Wilson, Sarang Gopalakrishnan, David A. Huse, and J. H. Pixley. Critical properties of the measurement-induced transition in random quantum circuits. Physical Review B, 101 (6): 060301, 2 2020. ISSN 24699969. 10.1103/PhysRevB.101.060301. https://doi.org/10.1103/PhysRevB.101.060301
[26] Anindita Bera and Sudipto Singha Roy. Growth of genuine multipartite entanglement in random unitary circuits. Physical Review A, 102 (6): 062431, 12 2020. ISSN 24699934. 10.1103/PhysRevA.102.062431. https://doi.org/10.1103/PhysRevA.102.062431
[27] Brayden Ware, Abhinav Deshpande, Dominik Hangleiter, Pradeep Niroula, Bill Fefferman, Alexey V. Gorshkov, and Michael J. Gullans. A sharp phase transition in linear cross-entropy benchmarking. 2023. https://doi.org/10.48550/arXiv.2305.04954. URL http://arxiv.org/abs/2305.04954. arXiv:2305.04954 [cond-mat, physics:quant-ph]. https://doi.org/10.48550/arXiv.2305.04954 arXiv:2305.04954
[28] Ramis Movassagh. Quantum supremacy and random circuits. 2019. https://doi.org/10.48550/arXiv.1909.06210. URL http://arxiv.org/abs/1909.06210. arXiv:1909.06210 [quant-ph]. https://doi.org/10.48550/arXiv.1909.06210 arXiv:1909.06210
[29] Alexander M. Dalzell, Nicholas Hunter-Jones, and Fernando G.S.L. Brandão. Random Quantum Circuits Anticoncentrate in Log Depth. PRX Quantum, 3 (1): 10333, 2022. ISSN 26913399. 10.1103/PRXQuantum.3.010333. https://doi.org/10.1103/PRXQuantum.3.010333
[30] Chi-Fang Chen, Jordan Docter, Michelle Xu, Adam Bouland, Fernando G.S.L. Brandão, and Patrick Hayden. Efficient unitary designs from random sums and permutations. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), pages 476–484, 2024. 10.1109/FOCS61266.2024.00037. https://doi.org/10.1109/FOCS61266.2024.00037
[31] Daniel Gottesman. The Heisenberg Representation of Quantum Computers. In S. P. Corney, R. Delbourgo, and P. D. Jarvis, editors, Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, pages 32–43, 1998. 10.48550/arXiv.quant-ph/9807006. https://doi.org/10.48550/arXiv.quant-ph/9807006 arXiv:quant-ph/9807006
[32] Sergey Bravyi and David Gosset. Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates. Physical Review Letters, 116 (25): 250501, 6 2016. ISSN 10797114. 10.1103/PhysRevLett.116.250501. https://doi.org/10.1103/PhysRevLett.116.250501
[33] Koenraad M R Audenaert and Martin B Plenio. Entanglement on mixed stabiliser states: Normal Forms and Reduction Procedures. New Journal of Physics, 7 (1): 170, 8 2005. 10.1088/1367-2630/7/1/170. https://doi.org/10.1088/1367-2630/7/1/170
[34] Otfried Gühne and Géza Tóth. Entanglement detection. Physics Reports, 474 (1-6): 1–75, 2009. ISSN 03701573. 10.1016/j.physrep.2009.02.004. https://doi.org/10.1016/j.physrep.2009.02.004
[35] Otfried Gühne and Michael Seevinck. Separability criteria for genuine multiparticle entanglement. New Journal of Physics, 12: 053002, 5 2010. ISSN 13672630. 10.1088/1367-2630/12/5/053002. https://doi.org/10.1088/1367-2630/12/5/053002
[36] Zhi Hao Ma, Zhi Hua Chen, Jing Ling Chen, Christoph Spengler, Andreas Gabriel, and Marcus Huber. Measure of genuine multipartite entanglement with computable lower bounds. Physical Review A - Atomic, Molecular, and Optical Physics, 83 (6): 062325, 6 2011. ISSN 10502947. 10.1103/PhysRevA.83.062325. https://doi.org/10.1103/PhysRevA.83.062325
[37] Liuke Lyu, Zi Yang Meng, Menghan Song, Ting-Tung Wang, and William Witczak-Krempa. Multiparty Entanglement Microscopy of Quantum Ising models in 1d, 2d and 3d. Physical Review B, 111 (24): 245108, 6 2025. 10.48550/arXiv.2412.12533. URL https://www.researchgate.net/publication/387141134. https://doi.org/10.48550/arXiv.2412.12533 https://www.researchgate.net/publication/387141134
[38] Adam Nahum and Brian Skinner. Entanglement and dynamics of diffusion-annihilation processes with Majorana defects. Physical Review Research, 2 (2): 023288, 6 2020. ISSN 26431564. 10.1103/PhysRevResearch.2.023288. https://doi.org/10.1103/PhysRevResearch.2.023288
[39] Nicolai Lang and Hans Peter Büchler. Entanglement Transition in the Projective Transverse Field Ising Model. Phys. Rev. B, 102 (9): 094204, 9 2020. 10.1103/PhysRevB.102.094204. URL http://arxiv.org/abs/2006.09748 http://dx.doi.org/10.1103/PhysRevB.102.094204. https://doi.org/10.1103/PhysRevB.102.094204 arXiv:2006.09748
[40] Ali Lavasani, Yahya Alavirad, and Maissam Barkeshli. Measurement-induced topological entanglement transitions in symmetric random quantum circuits. Nature Physics, 17: 342–347, 2021. 10.1038/s41567-020-01112-z. URL https://doi.org/10.1038/s41567-020-01112-z. https://doi.org/10.1038/s41567-020-01112-z
[41] 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, Feb 2021. 10.1103/PhysRevX.11.011030. URL https://link.aps.org/doi/10.1103/PhysRevX.11.011030. https://doi.org/10.1103/PhysRevX.11.011030
[42] Seung Ki Baek, Petter Minnhagen, and Beom Jun Kim. Surface and bulk criticality in midpoint percolation. Phys. Rev. E, 81: 041108, Apr 2010. 10.1103/PhysRevE.81.041108. URL https://link.aps.org/doi/10.1103/PhysRevE.81.041108. https://doi.org/10.1103/PhysRevE.81.041108
[43] Bastian Jungnitsch, Tobias Moroder, and Otfried Gühne. Taming multiparticle entanglement. Physical Review Letters, 106 (19): 190502, 5 2011. ISSN 00319007. 10.1103/PhysRevLett.106.190502. https://doi.org/10.1103/PhysRevLett.106.190502
[44] Tzu-Chieh Wei and Paul M. Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A, 68: 042307, Oct 2003. 10.1103/PhysRevA.68.042307. URL https://link.aps.org/doi/10.1103/PhysRevA.68.042307. https://doi.org/10.1103/PhysRevA.68.042307
[45] Oliver Lunt, Marcin Szyniszewski, and Arijeet Pal. Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional clifford circuits. Phys. Rev. B, 104: 155111, Oct 2021. 10.1103/PhysRevB.104.155111. URL https://link.aps.org/doi/10.1103/PhysRevB.104.155111. https://doi.org/10.1103/PhysRevB.104.155111
[46] Sergey Bravyi, David Fattal, and Daniel Gottesman. GHZ extraction yield for multipartite stabilizer states. Journal of Mathematical Physics, 47 (6): 062106, 6 2006. 10.1063/1.2203431. https://doi.org/10.1063/1.2203431
[47] John L Cardy. Critical Percolation in Finite Geometries. Journal of Physics A: Mathematical and General, 25 (4): L201–L206, 1992. 10.1088/0305-4470/25/4/009. https://doi.org/10.1088/0305-4470/25/4/009
[48] Miguel Navascués, Elie Wolfe, Denis Rosset, and Alejandro Pozas-Kerstjens. Genuine Network Multipartite Entanglement. Physical Review Letters, 125 (24): 240505, 12 2020. ISSN 10797114. 10.1103/PhysRevLett.125.240505. https://doi.org/10.1103/PhysRevLett.125.240505
[49] Arnau Lira-Solanilla, Xhek Turkeshi, and Silvia Pappalardi. Multipartite entanglement structure of monitored quantum circuits. Physical Review Letters, 135 (8): 080401, 8 2025. 10.1103/fl34-h1p1. URL http://arxiv.org/abs/2412.16062. https://doi.org/10.1103/fl34-h1p1 arXiv:2412.16062
[50] Yu Guo and Lizhong Huang. Complete monogamy of the multipartite quantum mutual information. Physical Review A, 107 (4): 042409, 4 2023. 10.1103/PhysRevA.107.042409. https://doi.org/10.1103/PhysRevA.107.042409
[51] Asutosh Kumar. Family of quantum mutual information in multiparty quantum systems. Physics Letters A, 529: 130091, 1 2025. https://doi.org/10.1016/j.physleta.2024.130091. https://doi.org/10.1016/j.physleta.2024.130091
[52] P Kleban, J J H Simmons, and R M Ziff. Anchored Critical Percolation Clusters and 2-D Electrostatics. Physical Review Letters, 97 (11): 115702, 9 2006. 10.1103/PhysRevLett.97.115702. https://doi.org/10.1103/PhysRevLett.97.115702
[53] David C Lewellen. Sewing constraints for conformal field theories on surfaces with boundaries. Nuclear Physics B, 372 (3): 654–682, 3 1992. 10.1016/0550-3213(92)90370-Q. https://doi.org/10.1016/0550-3213(92)90370-Q
[54] John L Cardy. Conformal Invariance and Surface Critical Behavior. Nuclear Physics B, 240 (4): 514–532, 6 1984. 10.1016/0550-3213(84)90241-4. https://doi.org/10.1016/0550-3213(84)90241-4
[55] Junfeng Wang, Zongzheng Zhou, Wei Zhang, Timothy M. Garoni, and Youjin Deng. Bond and Site Percolation in Three Dimensions. Physical Review E, 87 (5): 052107, 5 2013. 10.1103/PhysRevE.87.052107. URL http://arxiv.org/abs/1302.0421 http://dx.doi.org/10.1103/PhysRevE.87.052107. https://doi.org/10.1103/PhysRevE.87.052107 arXiv:1302.0421
[56] Simon Anders and Hans J. Briegel. Fast simulation of stabilizer circuits using a graph-state representation. Physical Review A, 73 (2): 022334, 2 2006. ISSN 10502947. 10.1103/PhysRevA.73.022334. https://doi.org/10.1103/PhysRevA.73.022334
[57] Liuke Lyu, James Allen, Yi Hong Teoh, Roger G Melko, and William Witczak-Krempa. Taming multiparty entanglement at measurement-induced phase transitions. 2026. 10.48550/arXiv.2602.04969. URL https://arxiv.org/abs/2602.04969. arXiv:2602.04969 [quant-ph]. https://doi.org/10.48550/arXiv.2602.04969 arXiv:2602.04969
[58] Shengqi Sang and Timothy H. Hsieh. Measurement Protected Quantum Phases. Physical Review Research, 3 (2): 023200–023209, 4 2020. 10.1103/PhysRevResearch.3.023200. URL http://arxiv.org/abs/2004.09509 http://dx.doi.org/10.1103/PhysRevResearch.3.023200. https://doi.org/10.1103/PhysRevResearch.3.023200 arXiv:2004.09509
[59] H. N. V. Temperley and Elliott H. Lieb. Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem. Proc. Roy. Soc. Lond. A, 322 (1549): 251–280, 4 1971. 10.1098/rspa.1971.0067. URL https://royalsocietypublishing.org/. https://doi.org/10.1098/rspa.1971.0067 https://royalsocietypublishing.org/
[60] W M Koo and H Saleur. Representations of the Virasoro algebra from lattice models. Nuclear Physics B, 426 (3): 459–504, 9 1994. 10.1016/0550-3213(94)90018-3. https://doi.org/10.1016/0550-3213(94)90018-3
[61] P. P. Martin. Temperley-Lieb algebra, group theory and the Potts model. J. Phys. A: Math. Gen, 21: 577, 2 1988. 10.1088/0305-4470/21/3/014. https://doi.org/10.1088/0305-4470/21/3/014
[62] J. Van Den Berg and H. Kesten. Inequalities with applications to percolation and reliability. Journal of Applied Probability, 22 (3): 556–569, 1985. 10.2307/3213860. https://doi.org/10.2307/3213860
[63] David Reimer. Proof of the van den Berg-Kesten conjecture. Combinatorics, Probability and Computing, 9 (1): 27–32, 2000. 10.1017/S0963548399004113. https://doi.org/10.1017/S0963548399004113
[64] Pieter W. Claeys, Marius Henry, Jamie Vicary, and Austen Lamacraft. Exact dynamics in dual-unitary quantum circuits with projective measurements. Physical Review Research, 4 (4): 043212, 12 2022. ISSN 26431564. 10.1103/PhysRevResearch.4.043212. https://doi.org/10.1103/PhysRevResearch.4.043212
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