The paper studies the time evolution of entanglement entropy in one-dimensional quantum systems, focusing on the transverse Ising spin chain. It explores how entanglement entropy evolves when the system starts from a pure state that is not an eigenstate of the Hamiltonian. Using path integral methods and explicit calculations, the authors find that entanglement entropy increases linearly with time up to a certain point, after which it saturates. This behavior is explained by causality, where entangled quasiparticles propagate at a maximum speed. In the critical case, the entanglement entropy scales with the length of the interval, with a coefficient depending on the central charge of the corresponding conformal field theory. For non-critical systems, the results are confirmed by exact calculations on the Ising model, showing that the entanglement entropy grows linearly with time until it saturates. The crossover time between linear and non-linear regimes is found to be $ t \sim \ell/2 $, consistent with the critical case. The paper also discusses the physical interpretation of these results, attributing the linear growth of entanglement entropy to the propagation of entangled quasiparticles. The results are validated by numerical diagonalization of the correlation matrix, showing that the entanglement entropy increases linearly with time and saturates at a value proportional to the interval length. The analysis reveals that the asymptotic value of the entanglement entropy depends on the initial and final magnetic fields, and that the results are consistent with conformal field theory predictions. The study highlights the importance of causality in understanding the time evolution of entanglement entropy in quantum systems.The paper studies the time evolution of entanglement entropy in one-dimensional quantum systems, focusing on the transverse Ising spin chain. It explores how entanglement entropy evolves when the system starts from a pure state that is not an eigenstate of the Hamiltonian. Using path integral methods and explicit calculations, the authors find that entanglement entropy increases linearly with time up to a certain point, after which it saturates. This behavior is explained by causality, where entangled quasiparticles propagate at a maximum speed. In the critical case, the entanglement entropy scales with the length of the interval, with a coefficient depending on the central charge of the corresponding conformal field theory. For non-critical systems, the results are confirmed by exact calculations on the Ising model, showing that the entanglement entropy grows linearly with time until it saturates. The crossover time between linear and non-linear regimes is found to be $ t \sim \ell/2 $, consistent with the critical case. The paper also discusses the physical interpretation of these results, attributing the linear growth of entanglement entropy to the propagation of entangled quasiparticles. The results are validated by numerical diagonalization of the correlation matrix, showing that the entanglement entropy increases linearly with time and saturates at a value proportional to the interval length. The analysis reveals that the asymptotic value of the entanglement entropy depends on the initial and final magnetic fields, and that the results are consistent with conformal field theory predictions. The study highlights the importance of causality in understanding the time evolution of entanglement entropy in quantum systems.