Measurement-only models offer an ideal platform for exploring entanglement dynamics in the absence of unitary evolution. Despite extensive numerical evidence for entanglement phase transitions in measurement-only dynamics, the underlying mechanism attributed to noncommutativity among multisite projective measurements has remained qualitative and coarse grained. In this work, we identify a quantitative noncommutative index for spatially and temporally homogeneous Pauli stabilizer measurement-only circuits. By applying this index to three representative measurement-only models, we find that the emergence of a volume-law phase is governed by the noncommutative structure of the measurement ensemble, while the transition point is quantitatively determined by the amount of critical noncommutativity. More strikingly, the critical noncommutativity exhibits a linear scaling with the measurement range, independent of the microscopic details of the measurement ensembles. Our findings deepen the understanding of the fundamental mechanism behind the measurement-only entanglement phase transition.

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• Phase transitions
• Quantum correlations, foundations & formalism
• Quantum entanglement
• Quantum measurements