Information geometry of sandwiched Rényi $α$-divergence

Abstract

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