Generalized entanglement entropies in two-dimensional conformal field theory

We introduce and study generalized Renyi entropies defined through the traces of products of Tr-B(vertical bar Psi(i)> are eigenstates of a two-dimensional conformal field theory (CFT). When vertical bar Psi(i)> = vertical bar Psi(j)> these objects reduce to the standard Renyi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Renyi entropies are equivalent to four point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Renyi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Renyi and relative entropies involving arbitrary descendent states of the bosonic CFT.