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arXiv:2609.28743

Classical Cellular Automaton for Measurement-Only Entanglement Transitions

Will Holdhusen, Bailey M. McAmis, Armin Rahmani

quant-phcond-mat.stat-mechSubmitted Sep 23, 2026• Announced Sep 25, 2026• 0 citations• Revised Oct 14, 2026 (v2)
v1 · Sep 23, 2026v2 · Oct 14, 2026
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Abstract

We introduce a generalized classical long-range stochastic cellular automaton that exactly captures the entanglement dynamics and transitions of a measurement-only monitored quantum system. The corresponding quantum model consists of a one-dimensional qubit chain subject to competing single-site and long-range Bell measurements, with separations drawn from a power-law distribution. At every step of each trajectory, the state remains a tensor product of Bell pairs and unpaired qubits, with bipartite entanglement entropy fully encoded in a classical matching of the sites. This exact representation allows us to solve the dynamics analytically in several limits and to determine the steady-state scaling of the entanglement entropy. We find volume-law, fractal, and area-law regimes. Numerical results show that these regimes persist beyond the analytically solvable limit. Because Bell measurement outcomes change only the Bell-state labels and leave the matching unchanged, all subsystem entropies are outcome independent, avoiding the exponential cost of trajectory postselection often encountered in experimental studies of measurement-induced entanglement dynamics.

Comments
5 pages, 5 figures, 5 pages of supplemental material

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We introduce a generalized classical long-range stochastic cellular automaton that exactly captures the entanglement dynamics and transitions of a measurement-only monitored quantum system. The corresponding quantum model consists of a one-dimensional qubit chain subject to competing single-site and long-range Bell measurements, with separations drawn from a power-law distribution. At every step of each trajectory, the state remains a tensor product of Bell pairs and unpaired qubits, with bipartite entanglement entropy fully encoded in a classical matching of the sites. This exact representation allows us to solve the dynamics analytically in several limits and to determine the steady-state scaling of the entanglement entropy. We find volume-law, fractal, and area-law regimes. Numerical results show that these regimes persist beyond the analytically solvable limit. Because Bell measurement outcomes change only the Bell-state labels and leave the matching unchanged, all subsystem entropies are outcome independent, avoiding the exponential cost of trajectory postselection often encountered in experimental studies of measurement-induced entanglement dynamics.
We investigate the entanglement dynamics of the Rule 54 quantum cellular automaton subject to stochastic projective measurements. Rule 54 is an interacting integrable model whose closed dynamics are exactly described by a semiclassical quasiparticle picture: entangled soliton pairs propagate ballistically and scatter elastically, generating entanglement at a rate determined analytically by the thermodynamic entropy density and the dressed quasiparticle velocity. We study how this picture is modified when measurements are interleaved with the unitary evolution, comparing two protocols that differ in their relationship to the conserved soliton charge. For measurements in the computational basis (Z-basis), which commute with the conserved charge, we derive an exact analytical prediction for the entanglement decay: the quasiparticle ansatz is modified by a measurement-induced exponential envelope exp(−4γt) , where γ is the measurement rate. This prediction is validated through finite-size scaling collapse across system sizes and measurement rates, with excellent agreement between analytics and Matrix Product State simulations. For measurements in the X-basis, which do not commute with the conserved charge, measurements act simultaneously as a source and a sink of entanglement through a quasiparticle reseeding mechanism. Our numerical results are consistent with a measurement-induced phase transition at a finite critical rate, in sharp contrast to the Z-basis case where any nonzero measurement rate drives the system to an area-law phase. A preliminary finite-size scaling analysis yields γc = 0.370 ± 0.004 (pc ≈ 0.31) and ν = 0.97 ± 0.03, though a complete characterization of the universality class requires larger system sizes.
Measurement-only circuits provide a minimal setting in which repeated local projections can either generate or suppress many-body entanglement, giving rise to measurement-induced phase transitions and dynamical regimes, that might have no unitary counterpart. Here we investigate entanglement and information transitions in one-dimensional measurement-only Clifford circuits with long-range two-qubit parity checks. By tuning both the measurement range and density per layer, we uncover a broad set of phases whose classification requires probes beyond entanglement entropy, such as mutual information, tripartite mutual information, purification from an ancilla, and Bell-cluster statistics. We map phase diagrams using large-scale Clifford simulations for two protocols: a random-basis design in which each measurement is randomly chosen from {XX,YY,ZZ}\lbrace XX,YY,ZZ \rbrace{XX,YY,ZZ}, and a single-basis design in which the basis is fixed within each layer but varies between layers, hence introducing more structure to the circuit. We map the trajectory-averaged entanglement entropy to a two-dimensional statistical mechanics model by extending a replica-based method to random-basis measurement-only circuits, and show that a continuous-time limit yields an effective long-range XX hamiltonian in the steady state. This connection links the observed volume-law to sub-volume-law entanglement transition to the boundary between a continuous symmetry broken phase and a critical XY phase. Strikingly, in structured (single-basis) circuits we find a phase in which volume-law and long-range entanglement coexists with rapid, size-independent purification of an ancilla qubit, and the absence of scrambling, highlighting measurement-only circuits as a promising route to efficiently preparing highly entangled and technologically useful quantum states.
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Measurement-induced entanglement phase transitions in monitored quantum circuits have stimulated activity in a diverse research community. However, the study of measurement-induced dynamics, due to the requirement of exponentially complex postselection, has been experimentally limited to small or specially designed systems that can be efficiently simulated classically. We present a solution to this outstanding problem by introducing a scalable protocol in U(1)U(1)U(1) symmetric circuits that facilitates the observation of entanglement phase transitions \emph{directly} from experimental data, without detailed assumptions of the underlying model or benchmarking with simulated data. Thus, the method is applicable to circuits which do not admit efficient classical simulation and allows a reconstruction of the full entanglement entropy curve with minimal theoretical input. Our approach relies on adaptive circuits and a steering protocol to approximate pure-state trajectories with mixed ensembles, from which one can efficiently filter out the subsystem U(1)U(1)U(1) charge fluctuations of the target trajectory to obtain its entanglement entropy. The steering protocol replaces the exponential costs of postselection and state tomography with a scalable overhead which, for fixed accuracy εεε and circuit size LLL, scales as Ns∼L5/2/ε\mathcal{N}_s\sim L^{5/2}/εNs​∼L5/2/ε.
The evolution of entanglement entropy in quantum circuits composed of Haar-random gates and projective measurements shows versatile behavior, with connections to phase transitions and complexity theory. We reformulate the problem in terms of a classical Markov process for the dynamics of bipartition purities and establish a probabilistic cellular-automaton algorithm to compute entanglement entropy in monitored random circuits on arbitrary graphs. In one dimension, we further relate the evolution of the entropy to a simple classical spin model that naturally generalizes a two-dimensional lattice percolation problem. We also establish a Markov model for the evolution of the zeroth Rényi entropy and demonstrate that, in one dimension and in the limit of large local dimension, it coincides with the corresponding second-Rényi-entropy model. Finally, we extend the Markovian description to a more general setting that incorporates continuous-time dynamics, defined by stochastic Hamiltonians and weak local measurements continuously monitoring the system.
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