Quantum Zeno effect and the many-body entanglement transition

We introduce and explore a one-dimensional “hybrid” quantum circuit model consisting of both unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and times throughout the system. By varying the measurement rate we can tune between the volume law entangled phase for the random unitary circuit model (no measurements) and a “quantum Zeno phase” where strong measurements suppress the entanglement growth to saturate in an area law. Extensive numerical simulations of the quantum trajectories of the many-particle wave functions (exploiting Clifford circuitry to access systems up to 512 qubits) provide evidence for a stable “weak measurement phase” that exhibits volume-law entanglement entropy, with a coefficient decreasing with increasing measurement rate. We also present evidence for a continuous quantum dynamical phase transition between the “weak measurement phase” and the “quantum Zeno phase,” driven by a competition between the entangling tendencies of unitary evolution and the disentangling tendencies of projective measurements. Detailed steady-state and dynamic critical properties of this quantum entanglement transition are accessed.

Figure 1

The structure of the hybrid circuit model. In this paper we will focus on 1D circuits with nearest neighbor gates. Each site has a spin-1/2 degree of freedom, and each block represents a gate operation on two qubits. See main text for details.

Figure 2

Results for model A1 [(a), (c), (e)] and model A2 [(b), (d), (f)]. (a), (b) The averaged steady-state value of the entanglement entropy for a system of size $L=16$ as a function of subsystem size $L_A$, for model A1 (a) and A2 (b), respectively. (a) There is almost no dependence of $S_A\left(L_A\right)$ on $L_A$, and this behavior strongly supports the area law of entanglement. (b) The behavior of $S_A\left(L_A\right)$ for model A2 is also consistent with an area law. (c), (d) The averaged steady-state entanglement entropy for different system sizes $L$, while keeping $L_A/L=1/2$ fixed, for model A1 (c) and A2 (d), respectively. (c) The half-system entanglement goes to a small constant as $L$ increases, again supporting the area law of entanglement. (d) The half-system entanglement appears to be subextensive in $L$. (e), (f) The steady state value of thermal entropy as a function of system size for $L$ up to 10 for A1 (e) and A2 (f), respectively. (e) The data points fit well on a straight line with slope 1. (f) The data points fit well on a straight line with slope 1.

Figure 3

The averaged entanglement entropy for models B1 and B2 are shown in panels (a) and (b), respectively, as a function of system size, for different values of $p$, on a log-log scale. All the data is taken with subsystem size $L_A=L/4$. In each figure one curve is highlighted with a thick line, corresponding to a critical value of $p=p_c$, that separates curves with $p<p_c$ that appear to asymptote to a straight line with slope $\approx 1$ at large $L$ (volume law), from the curves with $p>p_c$ which saturate to lines with slope 0 at large $L$ (area law).

Figure 4

Results for model B1, plotted on a semilog scale. In both panels, we take the subsystem size $L_A=L/4$. (a) The entanglement entropy as a function of $p$, for different system sizes. (b) $S_A/L^{\gamma }$ versus $(p-p_c)L^{1/\nu }$, for $0.05<p<0.3$. We find $p_c=0.15$, $\nu =1.85$, $\gamma =0.30$ for a best collapse.

Figure 5

Results for model B2, plotted on a semilog scale. In both panels, we take the subsystem size $L_A=L/4$. (a) The entanglement entropy as a function of $p$, for different system sizes. (b) $S_A/L^{\gamma }$ versus $(p-p_c)L^{1/\nu }$, for $0.3<p<1.0$. We find $p_c=0.68$, $\nu =1.75$, $\gamma =0.33$ for a best collapse.

Figure 6

$S_A{L}^{-\gamma }$ versus $t{L}^{-z}$ (where $z=1$) at criticality, $p=p_c$, in model B1. The entanglement entropy is computed for a subsystem of size $L_A=L/4$. The dashed line shows the function $f\left(x\right)\sim x^{\gamma /z}$, where $x=t{L}^{-z}$, which matches the data collapse well as $x\to 0$.