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 $S(\rho_1 \| \rho_0)$ 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 ${\rm Tr} ( \rho_1 \rho_0^{n-1} )$ and define a set of R\'enyi relative entropies $S_n(\rho_1 \| \rho_0)$. 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 $i \partial\phi$, 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.