Evolution of Entanglement Entropy in One-Dimensional Systems

TL;DR

This paper investigates how entanglement entropy evolves over time in one-dimensional quantum systems, revealing a linear increase up to a saturation point governed by signal propagation speed and initial state conditions.

Contribution

It combines quantum field theory path integral methods with explicit spin chain computations to analyze entanglement dynamics in 1D systems, highlighting causality constraints.

Findings

Entanglement entropy grows linearly with time until saturation.

Maximum signal propagation speed limits entanglement spread.

Saturation value depends on initial state and system size.

Abstract

We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length l and its complement, starting from a pure state which is not an eigenstate of the hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time $t$ up to t=l/2v, after which it saturates at a value proportional to l, the coefficient depending on the initial state. This behavior may be understood as a consequence of causality.