  We introduce and study generalized R\'enyi entropies defined through the
traces of products of ${\rm Tr}_B (|\Psi_i\rangle\langle \Psi_j|)$ where
$|\Psi_i\rangle$ are eigenstates of a two-dimensional conformal field theory
(CFT). When $|\Psi_i\rangle=|\Psi_j\rangle$ these objects reduce to the
standard R\'enyi entropies of the eigenstates of the CFT. Exploiting the path
integral formalism, we show that the second generalized R\'enyi 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 R\'enyi entropy for all eigenstates.
As a byproduct, our approach also leads to new results for the standard R\'enyi
and relative entropies involving arbitrary descendent states of the bosonic
CFT.
