Bounds on Rényi entropy growth in many-body quantum systems

We prove rigorous bounds on the growth of $\alpha$-Rényi entropies $S_{\alpha }\left(t\right)$ (the von Neumann entropy being the special case $\alpha =1$) associated with any subsystem $A$ of a general lattice quantum many-body system with finite on-site Hilbert space dimension. For completely nonlocal Hamiltonians, we show that the instantaneous growth rates $|S_{\alpha }^{\prime }\left(t\right)|$ (with $\alpha \ne 1$) can be exponentially larger than $|S_1^{\prime }\left(t\right)|$ as a function of the subsystem size $\left|A\right|$. For $D$-dimensional systems with geometric locality, we prove bounds on $|S_{\alpha }^{\prime }\left(t\right)|$ that depend on the decay rate of interactions with distance. When $\alpha =1$, the bound is $\left|A\right|$ independent for all power-law decaying interactions $V\left(r\right)\sim r^{-w}$ with $w>2D+1$. However, for $\alpha >1$, the bound is $\left|A\right|$ independent only when the interactions are finite-range or decay faster than $V\left(r\right)\sim e^{-c{r}^D}$ for some $c$ depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on $k$-local systems with or without geometric locality. A central theme of this work is that the value of $\alpha$ strongly influences the interplay between locality and instantaneous entanglement growth. In other words, the von Neumann entropy and the $\alpha$-Rényi entropies cannot be regarded as proxies for each other in studies of entanglement dynamics. We compare these bounds with analytic and numerical results on Hamiltonians with varying degrees of locality and find concrete examples that almost saturate the bound for nonlocal dynamics.

Figure 1

The red rectangle marks the boundary of the full lattice $\mathrm{\Xi }$, which we partition into $A,\overline{A}$ subsystems. For each lattice site $i$, the support of the interaction term $h_i\left(r\right)$ is contained in a ball of radius $r+1$ around $i$. ${\mathrm{\Lambda }}_A,{\mathrm{\Lambda }}_{\overline{A}}$ are the intersections of the support with $A,\overline{A}$, respectively. Note that the interaction only contributes to the entanglement rate when ${\mathrm{\Lambda }}_A,{\mathrm{\Lambda }}_{\overline{A}}$ are both nonempty. Moreover, the number of sites contained in the support is bounded by $O\left(r^d\right)$.

Figure 2

Time dependence of the von Neumann and second Rényi entropy for different values of $J$. Here, we chose units in which $J,t$ are both dimensionless. The dynamics are generated by the MFIM with $L=16$ and varying values of $J$. The entanglement entropy is calculated with respect to the middle cut. For each $J$, the plotted entropy growth curve is an average over 100 random product states. Note that, after an initial transient, the entropy grows linearly until saturation. Within the range $J\in [0.12,1.1]$, the slope of these curves grows monotonically with $J$.

Figure 3

$J$ dependence of the Von Neumann entropy growth rate $|S_1^{\prime }\left(t\right)|$ and the entanglement velocity $v_E$. Numerically, we take the curves in Fig. 2, and fit the intermediate time regime to a line (the early time oscillatory regime and the late time saturation regime are nonlinear and hence excluded from the fit). The slope of each curve is the growth rate $|S_1^{\prime }\left(t\right)|$ and the entanglement velocity can be extracted as $v_E=\left|S_1^{\prime }\left(t\right)\right|/(Jln2)$. The error bars for the growth rates are on the order of ${10}^{-3}–{10}^{-4}$ and hence invisible on the scale of the plot.