Probing prethermal nonergodicity through measurement outcomes of monitored quantum dynamics

Abstract

Projective measurements are a key element in quantum physics and enable rich phenomena in monitored quantum dynamics. Here, we show that the measurement outcomes, recorded during monitored dynamics, can provide crucial information about the properties of the monitored dynamical system itself. We demonstrate this for a Floquet model of many-body localization, where we find that the prethermal many-body localized regime becomes unstable against rare measurements, yielding an unusual enhancement of quantum entanglement. Through an unsupervised learning and mutual information analysis on the classical dataset of measurement outcomes, we find that the information loss in the system, reflected by the increased entanglement, is compensated by an emergent structure in this classical dataset. Our findings highlight the crucial role of measurements and corresponding classical outcomes in capturing prethermal nonergodicity, offering a promising perspective for applications to other monitored quantum dynamics.

Figures (4)

• Figure 1: (a) Schematic illustration of our monitored quantum circuit, whose unitary dynamics is governed by a Floquet model of MBL with gates $\hat{U}(\alpha)$ interspersed with stroboscopic projective measurements in the Z direction. We record the outcome of measurements during the dynamics of the quantum circuit in the form of a dataset, where each column represents the measurement outcomes of a single realization for the monitored quantum circuit. This dataset is then analyzed by means of principal component analysis (PCA) and mutual information. (b) For the system size $L=12$, the difference of the long-time entanglement entropy (EE) between the purely unitary evolution and the quantum circuit interspersed with rare measurements (at probability $p=10^{-4}$), i.e., $|\Delta \overline{S}| = |\overline{S}(p=10^{-4}) - \overline{S}(p=0)|$ as a function of disorder strength $\alpha$. Here, two vertical dashed lines highlight an estimation of the prethermal MBL regime with the lower boundary $\alpha \simeq 3$ and the upper boundary $\alpha \in [25,32]$ based on Ref. PhysRevB.105.174205. The inset shows the long-time EE for the unitary dynamics (dotted line) and with rare measurements (solid line) as a function of disorder strength $\alpha$.
• Figure 2: Comparison of the entanglement entropy (EE) dynamics for the quantum circuit in Fig. \ref{['fig1']}(a) with system size $L=12$ comparing purely unitary evolution (with measurement rate $p=0$) and rare measurements ($p=10^{-4}$) up to a final stroboscopic time $n=5\times 10^{4}$ for disorder strength $\alpha=10$ (a) and $\alpha=30$ (b). The number of realizations for the monitored quantum circuit is larger than $10^{3}$.
• Figure 3: (a) The PCA eigenvalues for the measurement dataset extracted from the dynamics of the quantum circuit with system size $L=12$ and three typical values of disorder strength $\alpha=3$, $10$, and $50$. (b) The values of mutual information between the $i-$th and $j-$th measurement outcome $I(i;j)$ for $\alpha=3$, $10$, and $50$. Here, we always consider rare measurements with a rate $p=10^{-4}$.
• Figure 4: (a) Growth of the temporal correlation length $\xi_{m}$ between measurements (normalized by the rate of measurements $p=10^{-4}$) as a function of the number of measurement $m$ for $\alpha=3$, $10$, and $50$. The dashed line shows the $\xi_{m}/p$ for the random dataset. (b) The slope $k$ of the correlation length growth between measurements as a function of disorder strength $\alpha$ compared to the PCA gap $\Delta$.