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arXiv:2511.08690 · Physical Review A

Fractal structure of multipartite entanglement in monitored quantum circuits

Vaibhav Sharma, Erich J Mueller

quant-phcond-mat.dis-nncond-mat.stat-mechSubmitted Nov 11, 2025• Announced Nov 13, 2025• 1 citation
v1 · Nov 11, 2025v2 · Jun 7, 2026
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Abstract

We analyze the distribution of multipartite entanglement in states produced in a one-dimensional random monitored quantum circuit where local Clifford unitaries are interspersed with single-site measurements performed with a probability $p$. This circuit has a measurement-induced phase transition at $p=p_c$, separating a phase in which the entanglement entropy scales with the system size (a volume law state) and one in which it scales with the boundary (an area law state). We calculate the entanglement depth, corresponding to the size of the largest cluster of entangled qubits, finding that it scales as a power law with system size in both the phases. The power law exponent is 1 in the volume law phase ($p < p_c$) and continuously decreases to 0 as $p \to 1$ in the area law phase. We explain this behavior by studying the spatial distribution of entangled clusters. We find that the largest cluster of entangled qubits in these states has a fractal dimension between 0 and 1 and appears to be self-similar. Away from the critical point, this fractal dimension matches the entanglement depth power law exponent.

Journal reference
Phys. Rev. A 113 (Editors' Suggestion), L060402 (2026)
DOI
10.1103/rdb3-gtwp
Comments
7 pages, 6 figures

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We analyze the distribution of multipartite entanglement in states produced in a one-dimensional random monitored quantum circuit where local Clifford unitaries are interspersed with single-site measurements performed with a probability ppp. This circuit has a measurement-induced phase transition at p=pcp=p_cp=pc​, separating a phase in which the entanglement entropy scales with the system size (a volume law state) and one in which it scales with the boundary (an area law state). We calculate the entanglement depth, corresponding to the size of the largest cluster of entangled qubits, finding that it scales as a power law with system size in both the phases. The power law exponent is 1 in the volume law phase (p<pcp < p_cp<pc​) and continuously decreases to 0 as p→1p \to 1p→1 in the area law phase. We explain this behavior by studying the spatial distribution of entangled clusters. We find that the largest cluster of entangled qubits in these states has a fractal dimension between 0 and 1 and appears to be self-similar. Away from the critical point, this fractal dimension matches the entanglement depth power law exponent.
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 ≥ 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.
Entanglement transitions in quantum dynamics present a novel class of phase transitions in non-equilibrium systems. When a many-body quantum system undergoes hybrid quantum dynamics, consisting of unitary evolution interspersed with monitored random measurements, the steady-state can exhibit a phase transition between volume- and area-law entanglement. The role of dimension in the nature of these transitions is an open problem. There is a dimensional correspondence between measurement-induced transitions in non-unitary quantum circuits in ddd spatial dimensions and classical statistical mechanical models in d+1d+1d+1 dimensions, where the time dimension in the quantum problem is mapped to a spatial dimension in the classical model. In this work we show that the role of dimension is considerably richer by unveiling a form of `dimensional hybridity': critical properties of the steady-state entanglement are governed by a combination of exponents consistent with ddd-dimensional percolation and (d+1)(d+1)(d+1)-dimensional percolation. We uncover this dimensional hybridity in 1+1D and 2+1D circuits using a graph-state based simulation algorithm where the entanglement structure is encoded in an underlying graph, providing access to the geometric structure of entanglement. We locate the critical point using the tripartite information, revealing area-law entanglement scaling at criticality, and showing that the entanglement transition coincides with the purification transition. The emergence of this `dimensional hybridity' in these non-unitary quantum circuits sheds new light on the universality of measurement-induced transitions, and opens the way for analyzing the quantum error correcting properties of random unitary circuits in higher dimensions.
The interplay of unitary evolution and projective measurements is a modern interest in the study of many-body entanglement. On the one hand, the competition between these two processes leads to the recently discovered measurement-induced phase transition (MIPT). On the other hand, measurement-based quantum computation (MBQC) is a well-known computational paradigm where measurements simulate unitary evolution by utilizing the entanglement of special resources such as the two-dimensional (2D) cluster state. The entanglement properties enabling MBQC may be attributed to symmetry-protected topological (SPT) orders, particularly subsystem-symmetry-protected topological (SSPT) orders. It was recently found that the one-dimensional cluster state may be associated with an SPT phase in random circuits respecting a global ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry, and furthermore that all phase transitions in this scenario belong to the same universality class. As resources with greater computational power feature greater symmetry, it is fruitful to investigate further any relationship between levels of symmetry in MIPTs and MBQC. In this paper we investigate MIPTs on a torus with three levels of symmetry-respecting unitary evolution interspersed by measurements. Although we find two area-law phases and one volume-law phase with distinct entanglement structures for each ensemble, the phase transition from the volume-law phase to the area-law phase associated with the 2D SSPT cluster state has variable correlation length exponent $\ensuremath{\nu}$. Whereas $\ensuremath{\nu}\ensuremath{\approx}0.90$ for unconstrained Clifford unitaries and $\ensuremath{\nu}\ensuremath{\approx}0.83$ for globally symmetric Cliffords, subsystem-symmetric Cliffords feature a much smaller value $\ensuremath{\nu}\ensuremath{\approx}0.38$. We discuss how these distinct $\ensuremath{\nu}$'s quantify spacetime response scales where quantum information is manipulated by single-qubit measurements as in MBQC.
Measurement-induced phase transitions (MIPTs) in monitored quantum circuits are usually characterized by preparing steady states at different uniform measurement probabilities. Here we introduce a spatial realization of the MIPT by imposing a deterministic measurement gradient in a single monitored Clifford chain. The resulting steady state contains coexisting volume-law, critical, and area-law regions, with the point p(x)=pcp(x)=p_cp(x)=pc​ acting as a spatial critical cut. By scanning entanglement observables across this profile, we show that the transition is organized by a spatial scaling form. Although this structure is analogous to finite-time scaling in temporally driven MIPT, the spatial protocol has no Kibble-Zurek dynamics. Instead, the physical bounds 0≤p≤10\le p\le 10≤p≤1 impose a finite linear window, producing cutoff-controlled asymptotic regimes whose fitted exponents provide direct access to the correlation-length exponent ννν. Our results establish spatially inhomogeneous measurements as a controlled route to engineer and probe measurement-induced criticality within a single steady state.
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We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent ααα. For α≲1α\lesssim 1α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size LLL is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2α\lesssim 1/2α≲1/2. For α>3/2α> 3/2α>3/2, the EE is in the area-law phase, namely, no scaling with LLL, for any monitoring strength. For 1<α≲3/21 < α\lesssim 3/21<α≲3/2, we identify an ααα-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.