Topological lower bound on quantum chaos by entanglement growth

TL;DR

This paper establishes a topological lower bound on entanglement growth in one-dimensional quantum cellular automata, linking it to the system's index and ruling out sublinear growth and many-body localization.

Contribution

It introduces a topological lower bound on entanglement growth in 1D QCA, connecting it to the index and providing a rigorous constraint on quantum chaos.

Findings

Lower bound on entanglement growth equals twice the QCA index.

Sublinear entanglement growth is impossible for nonzero index QCA.

Many-body localization is ruled out for unitaries with nonzero index.

Abstract

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