When the subsystem size is given by a half of the total space, we have investigated the time evolution of (Renyi) entanglement entropies for those locally excited states which are defined by acting local operator on the ground state. We have found that they approach finite constants in free field theories. We defined (Renyi) entanglement entropies of local operators by final values of those (Renyi) entanglement entropies. We have found that they depend on the details of local operators. We expect that they characterize local operators from the viewpoint of quantum entanglement. They help us study higher dimensional CFTs more. We also found the sum rule which those entropies obey. We also found that these results are interpreted in terms of the relativistic propagation of quasi-particles. We have investigated these quantities in strongly coupled theory. In this theory, it does not approach constants. It increases logarithmically with time. It is expected the late time behaviors of (Renyi) entanglement entropies characterize field theories from the viewpoint of quantum entanglement.