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arXiv:2603.12626

Critical behaviors of magic and participation entropy at measurement induced phase transitions

Eliot Heinrich, Hanchen Liu, Tianci Zhou, Xiao Chen

quant-phcond-mat.stat-mechSubmitted Mar 13, 2026• Announced Mar 16, 2026• 0 citations
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Abstract

We study the participation and stabilizer entropy of non-unitary quantum circuit dynamics, focusing on the critical line that separates the low-entanglement spin-glass phase and the paramagnetic phase. Along this critical line, the entanglement has a logarithmic scaling, which enables us to access the critical regime using large-scale matrix product state simulations with modest bond dimension. We find that both the participation entropy and stabilizer entropy exhibit critical slowing down: their saturation time scales linearly with the system size, in stark contrast to purely unitary dynamics, where saturation occurs on logarithmic time scales. In addition, we study bipartite participation and stabilizer mutual information, and find that it shows similar scaling behavior to the entanglement entropy. Finally, by analyzing the participation entropy of several paradigmatic Clifford circuits, we identify similar slow dynamical behavior near their respective critical points.

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We study the participation and stabilizer entropy of non-unitary quantum circuit dynamics, focusing on the critical line that separates the low-entanglement spin-glass phase and the paramagnetic phase. Along this critical line, the entanglement has a logarithmic scaling, which enables us to access the critical regime using large-scale matrix product state simulations with modest bond dimension. We find that both the participation entropy and stabilizer entropy exhibit critical slowing down: their saturation time scales linearly with the system size, in stark contrast to purely unitary dynamics, where saturation occurs on logarithmic time scales. In addition, we study bipartite participation and stabilizer mutual information, and find that it shows similar scaling behavior to the entanglement entropy. Finally, by analyzing the participation entropy of several paradigmatic Clifford circuits, we identify similar slow dynamical behavior near their respective critical points.
We investigate magic and its connection to entanglement in 1+1 dimensional random free fermion circuits, with a focus on hybrid free fermion dynamics that can exhibit an entanglement phase transition. To quantify magic, we use the Stabilizer Rényi Entropy (SRE), which we compute numerically via a perfect sampling algorithm. We show that although the SRE remains extensive as the system transitions from a critical phase to an area-law (disentangled) phase, the structure of magic itself undergoes a delocalization phase transition. This transition is characterized using the bipartite stabilizer mutual information, which exhibits the same scaling behavior as entanglement entropy: logarithmic scaling in the critical phase and a finite constant in the area-law phase. Additionally, we explore the dynamics of SRE. While the total SRE becomes extensive in O(1)O(1)O(1) time, we find that in the critical phase, the relaxation time to the steady-state value is parameterically longer than that in generic random circuits. The relaxation follows a universal form, with a relaxation time that grows linearly with the system size, providing further evidence for the critical nature of the phase.
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We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to L=48L=48L=48 for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.
Measurement-induced phase transition (MIPT) describes the nonanalytical change of the entanglement entropy resulting from the interplay between measurement and unitary evolution. In this paper, we investigate the relaxation critical dynamics near the MIPT for different initial states in a one-dimensional quantum circuit. Specifically, when the initial state is in the volume-law phase with vanishing measurement probability, we find that the half-chain entanglement entropy SSS decays as S∝t−1S\propto t^{-1}S∝t−1 with the coefficients proportional to the size of the system in the short-time stage; In contrast, when the initial state is the product state, SSS increases with time as S∝ln⁡tS\propto \ln{t}S∝lnt, consistent with previous studies. Despite these contrasting behaviors, we develop a unified scaling form to describe these scaling behaviors for different initial states where the off-critical-point effects can also be incorporated. This framework offers significant advantages for experimental MIPT detection. Our novel scheme, leveraging relaxation dynamical scaling, drastically reduces post-selection overhead, and can eliminate it completely with trackable classical simulation.
Random quantum circuits provide a framework for probing universal dynamical phenomena in chaotic quantum many-body systems, by circumventing the microscopic details of Hamiltonian evolution. Bipartite entanglement measures can be used to characterize the dynamical phases that emerge in these systems. Typically, under random unitary evolution, bipartite entanglement entropies exhibit a linear growth, and attain steady-state values that scale extensively with the system volume. However, when local projective measurements are introduced, a suppression of steady-state bipartite entropies occurs, leading to a sub-extensive scaling with the system volume. This class of entanglement scaling transitions, from an extensive to a sub-extensive scaling regime, has sparked interest in a broader class of measurement-induced phase transitions, with a potential to further our understanding of fault-tolerance in quantum computing, and in realizing synthetic quantum phases on digital quantum simulators. In this work, we characterize entanglement scaling transitions in one-dimensional random quantum circuits composed of q-level systems, with the local Hilbert space dimension q ≥ 2. This predominantly numerical study employs generalizations of the stabilizer formalism and the Clifford group to higher-dimensional Hilbert spaces to simulate the measurement-induced transitions. We find that the nature of the entanglement scaling transition remains invariant under changes in the local Hilbert space dimension. However, the critical measurement rate inducing the transition increases with q, asymptotically approaching 1/2 in the limit q → ∞. To further explore this classical limit, we develop a minimal dynamical model that captures entanglement structures under unitary evolution. We analyze the entanglement structures in thermal stabilizer states using this model, and propose an approach for incorporating measurements within this framework.
Quantum state complexity metrics, such as anticoncentration and nonstabilizerness, offer key insights into many-body physics, information scrambling, and quantum computing. Anticoncentration and equilibration of magic resources under dynamics of random quantum circuits occur at times scaling logarithmically with system size. Here, we examine these phenomena in one-dimensional ergodic Floquet models and thermalizing Hamiltonian systems. Using participation and stabilizer entropies to probe anticoncentration and magic resources, we reveal significant differences between the two settings. Floquet systems align with random circuit predictions, exhibiting anticoncentration and magic saturation at timescales logarithmic in system size. In contrast, Hamiltonian dynamics deviate from the random circuit predictions and require times scaling approximately linearly with system size to achieve saturation of participation and stabilizer entropies, which remain smaller than that of the typical quantum states even in the long-time limit. Our findings establish the phenomenology of participation and stabilizer entropy growth in generic many-body systems and emphasize the role of conservation laws in constraining anticoncentration and magic dynamics.