Information Geometry of the Geodesic Quantum $f$-Divergences

Abstract

We study the differential, statistical, and geometrical consequences generated by the geodesic quantum $f$-divergences introduced in [14], which are constructed by interpolating the relative modular operator and the commutant Radon-Nikodym derivative using a geodesic with parameter $t\in [0,1]$. For an invertible state $\rho$ and an operator convex function $f$, we compute the Hessian and obtain an explicit formula for the induced monotone quantum information metric $g_{\rho ,t}^{(f)}$. Furthermore, we also compare these metrics with the Petz-Hasegawa metric and find the meaning of the interpolation parameter $t$ in this new geometry. We next show that the interpolation of relative modular operators ${\Gamma }_t$ in the reference purification of a state $\sigma$ defines a canonical finite binary experiment $(p_t,q_t)$, and introduce a log-likelihood cumulant function ${\Psi }_{\rho ,\sigma }(t,s)$, recovering the Nussbaum-Szkoła distributions at $t=0$ and the Matsumoto construction at $t=1$. Finally, using Busemann functions, we endow this statistical framework with a geometric meaning in the cone of positive operators.