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 $S({\rho }_1\mathrm{\parallel }{\rho }_0)$ between two given reduced density matrices ${\rho }_1$ and ${\rho }_0$ of a quantum field theory, we employ the replica trick which relies on the path integral representation of ${\rm Tr} ( \rho_1 \rho_0^{n-1} )$anddefineasetofR\acute{e}nyirelativeentropies$S_n(\rho_1 | \rho_0)$\mathrm{.}Wecomputethesequantitiesforintegervaluesoftheparameter$n$ 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 $i\mathrm{\partial }\varphi$, 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.