  The relative entropy is a measure of the distinguishability of two quantum
states. A great deal of progress has been made in the study of the relative
entropy between an excited state and the vacuum state of a conformal field
theory (CFT) reduced to a spherical region. For example, when the excited state
is a small perturbation of the vacuum state, the relative entropy is known to
have a universal expression for \textit{all} CFT's
\cite{Faulk-GR-entanglement}. Specifically, the perturbative relative entropy
can be written as the symplectic flux of a certain scalar field in an
\textit{auxiliary} AdS-Rindler spacetime \cite{Faulk-GR-entanglement}.
Moreover, if the CFT has a semi-classical holographic dual, the relative
entropy is known to be related to conserved charges in the bulk dual spacetime
\cite{lashkari2016gravitational}. In this paper, we introduce a one-parameter
generalization of the relative entropy which we call \textit{refined} R\'enyi
relative entropy. We study this quantity in CFT's and find a one-parameter
generalization of the aforementioned known results about the relative entropy.
We also discuss a new family of positive energy theorems in asymptotically
locally AdS spacetimes that arises from the holographic dual of the refined
R\'enyi relative entropy.
