Topological Lower Bound on Quantum Chaos by Entanglement Growth

A fundamental result in modern quantum chaos theory is the Maldacena-Shenker-Stanford upper bound on growth of out-of-time-order correlators, whose infinite-temperature limit related to operator-space entanglement entropy evolution operator. Here we show that, for one-dimensional cellular automata (QCA), there exists a lower quantified by such entropy. This equal twice index QCA, which topological invariant that measures chirality information flow, and holds all R\'enyi entropies, with its strongest R\'enyi-$\infty$ version being tight. The rigorous rules out possibility any sublinear behavior, showing particular many-body localization forbidden unitary evolutions displaying nonzero index. Since measurable, our findings have direct experimental relevance. Our robust against exponential tails naturally appear dynamics generated local Hamiltonians.