This paper investigates the time evolution of entanglement entropy in one-dimensional systems, starting from a pure state that is not an eigenstate of the Hamiltonian. The authors use both path integral methods from quantum field theory and explicit computations for the transverse Ising spin chain. In both approaches, a maximum speed \( v \) for signal propagation is observed. The entanglement entropy generally increases linearly with time \( t \) up to \( t = \ell / 2v \), after which it saturates at a value proportional to \( \ell \), with the coefficient depending on the initial state. This behavior is attributed to causality. The paper also discusses the dynamics of entanglement entropy in the quantum Ising model, showing that for large times, the entanglement entropy is proportional to \( \ell \) for each pair of magnetic fields \( h \) and \( h_0 \). The results are consistent with conformal field theory predictions and provide insights into the evolution of entanglement in non-critical systems.This paper investigates the time evolution of entanglement entropy in one-dimensional systems, starting from a pure state that is not an eigenstate of the Hamiltonian. The authors use both path integral methods from quantum field theory and explicit computations for the transverse Ising spin chain. In both approaches, a maximum speed \( v \) for signal propagation is observed. The entanglement entropy generally increases linearly with time \( t \) up to \( t = \ell / 2v \), after which it saturates at a value proportional to \( \ell \), with the coefficient depending on the initial state. This behavior is attributed to causality. The paper also discusses the dynamics of entanglement entropy in the quantum Ising model, showing that for large times, the entanglement entropy is proportional to \( \ell \) for each pair of magnetic fields \( h \) and \( h_0 \). The results are consistent with conformal field theory predictions and provide insights into the evolution of entanglement in non-critical systems.