Short-time vs. long-time dynamics of entanglement in quantum lattice models

TL;DR

This paper investigates the short- and long-time dynamics of entanglement in quantum lattice models, deriving bounds for purity and entanglement growth, with implications for understanding quantum information spread.

Contribution

It introduces a new lower bound for the purity evolution in quantum lattice systems, linking entanglement dynamics to boundary size and providing a bridge between short- and long-time behaviors.

Findings

Purity decreases quadratically at short times, inversely proportional to boundary size.

An exponential lower bound for entanglement growth is established for larger times.

Numerical simulations confirm the theoretical bounds in specific quantum spin systems.

Abstract

We study the short-time evolution of the bipartite entanglement in quantum lattice systems with local interactions in terms of the purity of the reduced density matrix. A lower bound for the purity is derived in terms of the eigenvalue spread of the interaction Hamiltonian between the partitions. Starting from an initially separable state the purity decreases as $1 - (t/\tau)^2$, i.e. quadratically in time, with a characteristic time scale $\tau$ that is inversly proportional to the boundary size of the subsystem, i.e., as an area-law. For larger times an exponential lower bound is derived corresponding to the well-known linear-in-time bound of the entanglement entropy. The validity of the derived lower bound is illustrated by comparison to the exact dynamics of a 1D spin lattice system as well as a pair of coupled spin ladders obtained from numerical simulations.