Information geometry of sandwiched R\'enyi $\alpha$-divergence

TL;DR

This paper explores the geometric structure induced by the sandwiched Rényi α-divergence on quantum states, revealing conditions for monotonicity of the metric and dual flatness of the manifold.

Contribution

It characterizes the information geometric properties of the sandwiched Rényi divergence, including monotonicity regions and dual flatness at α=1, on quantum state spaces.

Findings

The Riemannian metric is monotone for α in (-∞, -1] ∪ [1/2, ∞).

The quantum statistical manifold is dually flat only at α=1.

The geometric structure varies significantly with α, impacting quantum information geometry.

Abstract

Information geometrical structure $(g^{(D_\alpha)}, \nabla^{(D_\alpha)},\nabla^{(D_\alpha)*})$ induced from the sandwiched R\'enyi $\alpha$-divergence $D_\alpha(\rho\|\sigma):=\frac{1}{\alpha (\alpha-1)}\log\,{\rm Tr} \left(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\,\sigma^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}$ on a finite quantum state space $\mathcal{S}$ is studied. It is shown that the Riemannian metric $g^{(D_\alpha)}$ is monotone if and only if $\alpha\in(-\infty, -1]\cup [\frac{1}{2},\infty)$, and that the quantum statistical manifold $({\mathcal{S}}, g^{(D_\alpha)}, \nabla^{(D_\alpha)},\nabla^{(D_\alpha)*})$ is dually flat if and only if $\alpha=1$.