Perturbative expansions of R\'enyi relative divergences and holography

Abstract

In this paper, we develop a novel way to perturbatively calculate R\'enyi relative divergences $D_{\gamma}(\rho|| \sigma) ={\rm tr} \rho^{\gamma} \sigma^{1-\gamma}$ and related quantities without using replica trick as well as analytic continuation. We explicitly determine the form of the perturbative term at any order by an integral along the modular flow of the unperturbed state. By applying the prescription to a class of reduced density matrices in conformal field theory, we find that the second order term of certain linear combination of the divergences has a holographic expression in terms of bulk symplectic form, which is a one parameter generalization of the statement "Fisher information = Bulk canonical energy".