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Spatial structure of multipartite entanglement at measurement induced phase transitions

James Allen and William Witczak-Krempa

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AbstractWe 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.Featured image: Entanglement-weighted graphs for 2- and 4-party entanglement in the measurement-only circuit.Popular summaryEntanglement, the key defining feature of the quantum world, is at its most interesting when many parties are entangled together at the same time and at long range, while it is at its most accessible when we can find it in small subregions without having to observe the whole system. Satisfying both these conditions is very hard, due to monogamy of entanglement – a system with many entangling interactions tends to look noisy on smaller scales, with entanglement between subregions decaying exponentially over distance or worse. However, if these interactions are moderated by measurements, multiparty entanglement on small subregions can become long-ranged – decaying with a power law over distance. Therefore, these non-unitary systems – typically random quantum circuits with a critical rate of measurements – have interesting and accessible entanglement properties. In this paper, we unravel the nature of the power law entanglement exponents in non-unitary systems. We develop a new model that accounts for the effect of measurements on long-range entanglement, and use it to argue that entanglement exponents should be subadditive – informally, the difficulty of generating higher-party entanglement should not increase more than linearly in the number of parties. We also look at a measurement-only circuit consisting of two competing types of measurements. This non-unitary system is connected to the percolation model, and we prove that all entanglement exponents are equal to twice the party number. This result, supported by numerical evidence up to 5 parties, is the first analytical expression for entanglement exponents in any non-unitary system.► BibTeX data@article{Allen2026spatialstructureof, doi = {10.22331/q-2026-09-18-2211}, url = {https://doi.org/10.22331/q-2026-09-18-2211}, title = {Spatial structure of multipartite entanglement at measurement induced phase transitions}, author = {Allen, James and Witczak-Krempa, William}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2211}, month = sep, year = {2026} }► References [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.

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Physical Review Research, 4 (4): 043212, 12 2022. ISSN 26431564. 10.1103/PhysRevResearch.4.043212. https://doi.org/10.1103/PhysRevResearch.4.043212Cited byCould not fetch Crossref cited-by data during last attempt 2026-09-18 10:19:14: Could not fetch cited-by data for 10.22331/q-2026-09-18-2211 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-09-18 10:19:14: 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. AbstractWe 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.Featured image: Entanglement-weighted graphs for 2- and 4-party entanglement in the measurement-only circuit.Popular summaryEntanglement, the key defining feature of the quantum world, is at its most interesting when many parties are entangled together at the same time and at long range, while it is at its most accessible when we can find it in small subregions without having to observe the whole system. Satisfying both these conditions is very hard, due to monogamy of entanglement – a system with many entangling interactions tends to look noisy on smaller scales, with entanglement between subregions decaying exponentially over distance or worse. However, if these interactions are moderated by measurements, multiparty entanglement on small subregions can become long-ranged – decaying with a power law over distance. Therefore, these non-unitary systems – typically random quantum circuits with a critical rate of measurements – have interesting and accessible entanglement properties. In this paper, we unravel the nature of the power law entanglement exponents in non-unitary systems. We develop a new model that accounts for the effect of measurements on long-range entanglement, and use it to argue that entanglement exponents should be subadditive – informally, the difficulty of generating higher-party entanglement should not increase more than linearly in the number of parties. We also look at a measurement-only circuit consisting of two competing types of measurements. This non-unitary system is connected to the percolation model, and we prove that all entanglement exponents are equal to twice the party number. This result, supported by numerical evidence up to 5 parties, is the first analytical expression for entanglement exponents in any non-unitary system.► BibTeX data@article{Allen2026spatialstructureof, doi = {10.22331/q-2026-09-18-2211}, url = {https://doi.org/10.22331/q-2026-09-18-2211}, title = {Spatial structure of multipartite entanglement at measurement induced phase transitions}, author = {Allen, James and Witczak-Krempa, William}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2211}, month = sep, year = {2026} }► References [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.

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Spatial structure of multipartite entanglement at measurement induced phase transitions

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