Information geometry of sandwiched Rényi $\alpha$-divergence

Abstract

Information geometrical structure $(g^{(D_{\alpha })},{\nabla }^{(D_{\alpha })},{\nabla }^{(D_{\alpha })\ast })$ induced from the sandwiched Rényi $\alpha$-divergence $D_{\alpha }(\rho \Vert \sigma ):=\frac{1}{\alpha (\alpha -1)}\log \mathrm{Tr}{\left({\sigma }^{\frac{1-\alpha }{2\alpha }}\rho {\sigma }^{\frac{1-\alpha }{2\alpha }}\right)}^{\alpha }$ 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 $(\mathcal{S},g^{(D_{\alpha })},{\nabla }^{(D_{\alpha })},{\nabla }^{(D_{\alpha })\ast })$ is dually flat if and only if $\alpha =1$.