Notes on Quantum Entanglement of Local Operators

This is an expanded version of the short report arXiv:1401.0539, where we stud- ied the (Renyi) entanglement entropies for the excited state defined by acting a given local operator on the ground state. We introduced the (Renyi) entanglement entropies of given local operators which measure the degrees of freedom of local operators and characterize them in conformal field theories from the viewpoint of quantum entanglement. In present paper, we explain how to compute them in free massless scalar field theories and we also investigate their time evolution. The results are interpreted in terms of relativistic propagation of an entangled pair. The main new results which we acquire in the present paper are as follows. Firstly, we provide an explanation which shows that the (Renyi) entanglement entropies of a specific operator are given by (Renyi) entanglement entropies of binomial distribution by the replica method. That operator is constructed of only scalar field. Secondly, we found the sum rule which (Renyi) entanglement entropies of those local operators obey. Those local operators are located separately. Moreover we argue that (Renyi) entanglement entropies of specific operators in conformal field theories are given by (Renyi) entanglement entropies of binomial distribution. These specific operators are constructed of single-species operator. We also argue that general operators obey the sum rule which we mentioned above.