The holographic entropy zoo

We study the holographic dual of a two parameter family of quantities known as the $\alpha$-$z$ divergences. Many familiar information theoretic quantities occur within this family, including the relative entropy, fidelity, and collision relative entropy. We find explicit bulk expressions for the boundary divergences to second order in a state perturbation whenever $\alpha$ is an integer and $z\geq0$, as well as when $z\in\{0,\infty\}$ and $\alpha\in \mathbb{R}$. Our results apply for perturbations around an arbitrary background state and in any dimension, under the assumption of the equality of bulk and boundary modular flows.