Relative entanglement entropies in 1 + 1-dimensional conformal field theories

We study the relative entanglement entropies of one interval between excited states of a 1+1 dimensional conformal field theory (CFT). To compute the relative entropy S(ρ1‖ρ0) between two given reduced density matrices ρ1 and ρ0 of a quantum field theory, we employ the replica trick which relies on the path integral representation of Tr(ρ1ρn−10) and define a set of R\'enyi relative entropies Sn(ρ1‖ρ0). We compute these quantities for integer values of the parameter n and derive via the replica limit, the relative entropy between excited states generated by primary fields of a free massless bosonic field. In particular, we provide the relative entanglement entropy of the state described by the primary operator i∂ϕ, both with respect to the ground state and to the state generated by chiral vertex operators. These predictions are tested against exact numerical calculations in the XX spin-chain finding perfect agreement. © 2017, The Author(s)