The dynamical $\alpha$-R\'enyi entropies of local Hamiltonians grow at most linearly in time

We consider a generic one dimensional spin system of length $ L $, arbitrarily large, with strictly local interactions, for example nearest neighbor, and prove that the dynamical $ \alpha $-R\'enyi entropies, $ 0 < \alpha \le 1 $, of an initial product state grow at most linearly in time. This result arises from a general relation among dynamical $ \alpha $-R\'enyi entropies and Lieb-Robinson bounds. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of $\alpha$ close enough to $ 1 $, and moreover to initial pure states with low entanglement, of order $ \log L $, that are typically represented by critical states. We establish that low entanglement states have an efficient MPS representation that persists at least up to times of order $ \log L $. The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on $ \alpha $-R\'enyi entropies and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest.