Measurement-induced entanglement transitions in the quantum Ising chain: From infinite to zero clicks

We investigate measurement-induced phase transitions in the quantum Ising chain coupled to a monitoring environment. We compare two different limits of the measurement problem: the stochastic quantum-state diffusion protocol corresponding to infinite small jumps per unit of time and the no-click limit, corresponding to postselection and described by a non-Hermitian Hamiltonian. In both cases we find a remarkably similar phenomenology as the measurement strength $\gamma$ is increased, namely, a sharp transition from a critical phase with logarithmic scaling of the entanglement to an area-law phase, which occurs at the same value of the measurement rate in the two protocols. An effective central charge, extracted from the logarithmic scaling of the entanglement, vanishes continuously at the common transition point, although with different critical behavior possibly suggesting different universality classes for the two protocols. We interpret the central charge mismatch near the transition in terms of noise-induced disentanglement, as suggested by the entanglement statistics which displays emergent bimodality upon approaching the critical point. The non-Hermitian Hamiltonian and its associated subradiance spectral transition provide a natural framework to understand both the extended critical phase, emerging here for a model which lacks any continuous symmetry, and the entanglement transition into the area law.

Figure 1

Cartoon of the monitored quantum Ising chain. (a) Quantum state diffusion protocol: on each lattice site a two-level system (Ising spin) interacts with its neighbors and is subject to a weak-continuous measurement of the up-polarized state. (b) Quantum jump protocol: the spins in the chain interact through a non-Hermitian Hamiltonian. Occasionally a quantum jump takes place, projecting the measured degree of freedom in the up-polarized state. The no-click limit corresponds to postselecting only trajectories without jumps (see text for details).

Figure 2

Dynamics of the entanglement entropy averaged over $N=2000$ trajectories for different values of the measurement strength $\gamma$, different system sizes $L$, and fixing $L_A=L/4$. (a)–(c) We see a logarithmic growth in time of the entanglement (note the logarithmic scale for the time axis) with a saturation to a size-dependent plateau, for $\gamma <4$ and a much slower growth to a value independent from system size, for $\gamma >4$.

Figure 3

Entanglement dynamics for the non-Hermitian Hamiltonian for different values of the measurement rate $\gamma$ and different system sizes. For small $\gamma$ [panel (a)] we see a logarithmic growth in time and a saturation to a plateau that depends on system sizes. For large $\gamma$ [panel (b)] the entanglement rapidly saturates to a value that does not depend on size. Approaching the transition point we see strong oscillations in the dynamics.

Figure 4

Stationary-state entanglement entropy as a function of the length of the chain. (a) Average entanglement entropy for the QSD protocol. (b) Long-time entanglement entropy for the non-Hermitian dynamics. In both cases we see a transition from a logarithmic critical scaling, for small $\gamma$ to an area law (constant) scaling at large $\gamma$. Different curves in the two panels correspond to increasing values of $\gamma$, from $\gamma =0.25$ (top curve) to $\gamma =6$ (bottom curve) with a $\mathrm{\Delta }\gamma =0.25$.

Figure 5

Effective central charge as a function of the measurement rate $\gamma$, for the QSD protocol and the non-Hermitian dynamics. In both cases we find that $c\left(\gamma \right)$ decreases monotonously and vanishes around ${\gamma }_c\simeq 4$, which locates the entanglement transition between logarithmic scaling and area law. Interestingly the two central charges coincide at small and large values of $\gamma$, while approaching the critical point we find the non-Hermitian one to be larger, possibly vanishing at ${\gamma }_c$ in a nonanalytic way.

Figure 6

(a)–(f) Probability distribution of the entanglement entropy for different values of $\gamma$ and different system sizes. The realization collected for these functions is $N=64000$. We see that upon increasing $\gamma$, the statistic evolves from a simple Gaussian [panels (a) and (b)] to a skewed distribution [panels (c)–(e)], with emergent bimodality. At large $\gamma$, the distribution is peaked at $S_{\infty }\simeq 0$ [panel (f)]. The solid line shows the average (blue line) over the trajectories for the quantum state diffusion, and the stationary value of the non-Hermitian Hamiltonian (orange line). (g),(h) Different statistical estimators $\langle \langle S\rangle \rangle$ versus system size for different values of $\gamma$. Note the logarithmic scaling in the $L$ axis.