Entanglement Growth from Entangled States: A Unified Perspective on Entanglement Generation and Transport

TL;DR

This work addresses how entanglement propagates in quantum many-body systems when starting from states that already possess substantial entanglement. It introduces a unified build–move framework to separate entanglement generation from transport and analyzes both Hamiltonian and random circuit dynamics, using the random SWAP circuit as a move baseline. A key finding is that, in many non-ergodic settings such as MBL, the growth of half-chain entanglement entropy is non-monotonic in the initial entanglement, revealing a move-dominated mechanism that redistributes pre-existing entanglement rather than simply creating new entanglement. The results provide a universal classification of entanglement dynamics and offer experimentally-testable predictions for how entanglement reservoirs are redistributed in complex quantum systems.

Abstract

Studies of entanglement dynamics in quantum many-body systems have focused largely on initial product states. Here, we investigate the far richer dynamics from initial entangled states, uncovering universal patterns across diverse systems ranging from many-body localization (MBL) to random quantum circuits. Our central finding is that the growth of entanglement entropy can exhibit a non-monotonic dependence on the initial entanglement in many non-ergodic systems, peaking for moderately entangled initial states. To understand this phenomenon, we introduce a conceptual framework that decomposes entanglement growth into two mechanisms: ``build'' and ``move''. The ``build'' mechanism creates new entanglement, while the ``move'' mechanism redistributes pre-existing entanglement throughout the system. We model a pure ``move'' dynamics with a random SWAP circuit, showing it uniformly distributes entanglement across all bipartitions. We find that MBL dynamics are ``move-dominated'', which naturally explains the observed non-monotonicity of the entanglement growth. This ``build-move'' framework offers a unified perspective for classifying diverse physical dynamics, deepening our understanding of entanglement propagation and information processing in quantum many-body systems.

Figures (8)

• Figure 1: The "build-move" framework and the increase of HCEE in various dynamics. (a-c) Schematic illustrations of "build" mechanism (a), "move" mechanism (b), and RQC composed of two-qubit gates $U$ (c). Blue circles represent spins, and pairs of spins linked by red double arrows indicate Bell pairs. (d,e) Classification of dynamics based on the HCEE growth: $\Delta S = S_{\text{sat}} - S_{\text{initial}}$, as a function of the initial HCEE, $S_{\text{initial}}$. All data points are for system size $L=16$. $S_{\text{initial}}$ is controlled by varying the preparation time $T$. (d) Results for Hamiltonian-based dynamics. (e) Results for RQC-based dynamics. The two-qubit gates $U$ are defined in Eq. (\ref{['eq:U_RQC']}), with parameters in various classes detailed in Table \ref{['tab:1']}. For the calculation details of the data in all the figures in the main text, please refer to the Supplemental Material Sec. \ref{['sec:numerical_details']}.
• Figure 2: Time evolution of HCEE $S$ with initial state $|\psi(T)\rangle$ for $L=16$ and $T=4.5$. (a) Evolution under localized dynamics. The lower axis shows time $t$ for AL and Hamiltonian MBL evolutions, while the upper axis shows the period number for the Floquet MBL dynamics. To visualize the dynamics across multiple timescales, the horizontal axis uses a hybrid scale of linear for the initial evolution ($t,\text{period number}<10$) and logarithmic for long times. (b) Evolution under RQCs dynamics. The horizontal axis represents circuit depth. The specific $(\alpha, \beta)$ values shown in the legend are the parameters of the quantum gates selected from their respective categories.
• Figure 3: Evolution of BAEE $\bar{S}$, HCEE $S$ and their difference under $\hat{H}(W=0.5)$ for $L=16$, starting from the product state $|\psi_0\rangle$. The data point corresponding to the maximum value of $\bar{S}-S$ is at $T=3.0$ with HCEE $S\approx1.34$, roughly coinciding with the $S_{\text{initial}}$ for the peak $\Delta S$ of move-dominated dynamics in Fig. \ref{['fig1']}(d,e).
• Figure S1: Level spacing ratios distribution $p(r)$ of the Hamiltonian $\hat{H}$ (Eq. \ref{['eqS1']}) for $L=16$. For each panel, the distribution is computed from the middle one-third of the eigenenergies in the half-filling sector, with data compiled from 100 independent disorder realizations for the corresponding disorder strength $W$. Solid lines represent the theoretical predictions from RMT. (a) Thermal phase ($W=0.5$). The distribution clearly follows the Wigner-Dyson form for the GOE. The numerically obtained average value of $\left\lbrace r_n\right\rbrace$ is about $0.531$, is in excellent agreement with the GOE prediction $\bar{r}=4-2\sqrt3\approx0.536$PhysRevLett.110.084101. (b) MBL phase ($W=5.0$). The distribution is well-described by Poisson statistics. The numerical average value of $\left\lbrace r_n\right\rbrace$ is about $0.388$, closely matches the theoretical value for a Poisson distribution, $\bar{r}=2\ln2-1\approx0.386$PhysRevLett.110.084101.
• Figure S2: Time evolution of HCEE $S$ with initial state $|\psi'(T)\rangle$ for $L=16$ and $T=3.0$. The lower axis shows time $t$ for the Hamiltonian MBL evolution, while the upper axis shows the period number for the Floquet MBL dynamics. To visualize the dynamics across multiple timescales, the horizontal axis uses a hybrid scale: it is linear for the initial evolution ($t,\text{period number}<10$) and becomes logarithmic for long times.