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 < 1, of an initial product state grow at most linearly in time. 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. We provide a non rigorous argument to extend our results to initial low-entangled, meaning O(log L) states. This class of states includes many examples of spin systems ground states, and also critical states. This implies 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.