Universal learning of nonlocal entropy via local correlations in non-equilibrium quantum states

TL;DR

This Letter employs a multilayer perceptron (MLP) to establish a universal mapping between the QMI and local correlations only up to second order for nonequilibrium states generated by quenches in a one-dimensional disordered XXZ model, providing a practical method for experimentally extracting QMI.

Abstract

Characterizing the nonlocal nature of quantum states is a central challenge in the practical application of large-scale quantum computation and simulation. Quantum mutual information (QMI), a fundamental nonlocal measure, plays a key role in quantifying entanglement and has become increasingly important in studying nonequilibrium quantum many-body phenomena, such as many-body localization and thermalization. However, experimental measurement of QMI remains extremely difficult, particularly for nonequilibrium states, which are more complex than ground states. In this Letter, we employ a multilayer perceptron (MLP) to establish a universal mapping between the QMI and local correlations only up to second order for nonequilibrium states generated by quenches in a one-dimensional disordered XXZ model. Our approach provides a practical method for experimentally extracting QMI, readily applicable in platforms such as superconducting qubits. Moreover, this work will establishes a general framework for reconstructing other nonlocal observables, including Fisher information and out-of-time-ordered correlators.

Figures (4)

• Figure 1: The protocol to training the Rény entropy: a. The quenching dynamics of XXZ model and generation of local correlation and QMI for non-equilibrium state; $\textbf{b}$. The MLP method to learn the non-linear mapping between QMI $\delta S^{(n)}$ and local irreducible entropy $\delta S_{i}^{(n)}$, $\delta S_{\{i,j\}}^{(n)}$ with $i,j$ be the index of the spins.
• Figure 2: Machine learning of the dynamics of QMI (a), (b), (c): The dynamics of QMI $\delta S^{(2)}/N$ of single disorder realization for different disorder strength: (a). $W=1$, (b). $W=3$, (c). $W=8$ when fixing $J_{z}=1$ and $N=10$. $\delta S^{(2)}$ is calculated for the partitions $A=\{1,2,3,4,5\}$ and $B=\{6,7,8,9,10\}$.(d), (e), (f): The disorder averaged dynamics of $\delta S^{(n)}/N$ upper corresponding to the case (e), (d), (f). These results are obtained by averaging over $100$ random disorder configuration. In all these figures, the circles, triangle and dashed line denotes the result that calculated by exact simulation, MLP and CCE method respectively.
• Figure 3: Universal learning of the long-time dynamics of QMI: The prediction of long-time dynamics of QMI (a). $\delta S^{(1)}$; (b). $\delta S^{(2)}$ under different disorder strength $W$ and anisotropy $J_{z}$ using the universal model trained under the parameters $W=1,J_{z}=2$ and quenching time $t\in[0,200]$. In all these figures, the solid and empty scatters denotes the the result of exact simulation and prediction by MLP respectively. Different colors and shape denotes different parameters as shown in the legend of the graphs. (c) and (d) shows the prediction of long-time dynamics of QMI $\delta S^{(1)}$ and $\delta S^{(2)}$ respectively for fixed time $t=10^{9}$ and anisotropy $J_{z}=2$ as a function of disorder strength $W$ using the universal model trained at the parameters $W=1,J_{z}=2$. The circle scatters and rectangle denotes the results from exact simulation and MLP respectively. The size of the system is taken to be $N=10$ and initial state to be Neel states.
• Figure 4: Robustness of the MLP method: (a) $N=9$, (b) $N=10$, (c) $N=11$ and $N=12$: Machine learning of of long time dynamic of the QMI for different strength of the measurement noise and different system size; (e) and (f): Prediction of QMI $\delta S^{(1)}$ and $\delta S^{(2)}$for fixed quenching time $t=10^{9}$ as a function of disorder strength $W$ under different noise strength $\sigma$. The size of the system is $N=10$. All these prediction use the model trained at $W=1,J_{z}=1$.