Non-commutative Index of Measurement-only Entanglement Phase Transition

Abstract

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 non-commutativity among multi-site projective measurements has remained qualitative and coarse-grained. In this work, we identify a quantitative non-commutative index. By applying this index into three representative measurement-only models, we elucidate the role of non-commutativity in measurement-only dynamics: the emergence of a volume-law phase is governed by the non-commutative structure of the measurement ensemble, while the transition point is quantitatively determined by the amount of critical non-commutativity. More strikingly, the critical non-commutativity exhibits a universal 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.

Figures (11)

• Figure 1: Circuit schematics for unitary, unitary-projective, and measurement-only dynamics. (a) MBL model represented as a quantum circuit, composed of three qualitatively distinct unitary gates corresponding to the three terms in the MBL Hamiltonian. (b) MIPT circuit in a brick-wall structure of random unitary gates interspersed with local-Z measurements applied at random sites. (c) Measurement-only circuit, consisting solely of joint multi-site measurement operations.
• Figure 2: Time evolution of the half-chain entanglement entropy $S_{1/2}$ along single trajectories for the factorizable measurement ensemble, starting from a pure zero state. Different curves correspond to different values of the probability parameter $q$, with the constraint $p_x=p_y=q$ and $p_z=1-2q$. The blue curve ($q=0$) represents the fully commuting case, where the entanglement entropy remains zero at all times and the steady state is area-law. For $q=0.10$ (orange), non-commuting measurements are present but the system still relaxes to an area-law steady state. At $q=0.13$ (green), close to the critical point, the entropy exhibits a slow growth characteristic of critical behavior. For $q=0.33$ (red), the entropy grows linearly at early times and saturates at a value proportional to the system size, indicating a volume-law phase.
• Figure 3: Non-commutativity index $\mathcal{I}(\mathcal{E})$ of the factorizable measurement ensembles with different measurement ranges $r$, plotted along two representative directions in probability space. (a) Symmetric line $p_X=p_Y=q_0$ and $p_Z=1-2q_0$. (b)Marginal line $p_X=q_0, p_Y=1-q_0$ and $p_Z=0$. In both cases, $\mathcal{I}(\mathcal{E})$ exhibits a single maximum, located at the maximally symmetric points $p_X=p_Y=p_Z=1/3$ in (a) and $p_X=p_Y=1/2, p_Z=0$ in (b), respectively. The horizontal lines indicate the respectively needed $\mathcal{I}$ for phase transition and the corresponding vertical lines depict the phase transition point.
• Figure 4: Critical non-commutativity $\mathcal{I}(\mathcal{E})$ of factorizable measurement ensembles evaluated at the entanglement phase transition along the symmetric line $p_X=p_Y=q_0$ for different interaction ranges $r$. The transition points $q_{0,critical}$ are determined from the analytical phase boundary given in Eq. (6).
• Figure 5: Critical non-commutativity $\mathcal{I}_c$ for factorizable ensembles extracted along different directions in probability space, including one edge of the parameter space and different 6 center-to-boundary lines of the parameter space. All data collapse onto the same linear $\mathcal{I}_c(r)$, same as Fig.3