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We prove rigorous bounds on the growth of $\alpha$-Renyi entropies$S_{\alpha}(t)$ (the Von Neumann entropy being the special case $\alpha = 1$)associated with any subsystem $A$ of a general lattice quantum many-body systemwith finite onsite Hilbert space dimension. For completely non-localHamiltonians, we show that the instantaneous growth rates $|S'_{\alpha}(t)|$(with $\alpha \neq 1$) can be exponentially larger than $|S'_1(t)|$ as afunction of the subsystem size $|A|$. For $D$-dimensional systems withgeometric locality, we prove bounds on $|S'_{\alpha}(t)|$ that depend on thedecay rate of interactions with distance. When $\alpha = 1$, the bound is$|A|$-independent for all power-law decaying interactions $V(r) \sim r^{-w}$with $w > 2D+1$. But for $\alpha > 1$, the bound is $|A|$-independent only whenthe interactions are finite-range or decay faster than $V(r) \sim e^{- c\,r^D}$ for some $c$ depending on the local Hilbert space dimension. Usingsimilar arguments, we also prove bounds on $k$-local systems with or withoutgeometric locality. A central theme of this work is that the value of $\alpha$strongly influences the interplay between locality and entanglement growth. Inother words, the Von Neumann entropy and the $\alpha$-Renyi entropies cannot beregarded as proxies for each other in studies of entanglement dynamics. Wecompare these bounds with analytic and numerical results on Hamiltonians withvarying degrees of locality and find concrete examples that almost saturate thebound for non-local dynamics.

Bounds on Renyi entropy growth in many-body quantum systems