Information Geometry of the Geodesic Quantum $f$-Divergences

• URL: http://arxiv.org/abs/2608.15916v1
• Date: Sun, 16 Aug 2026 20:14:02 GMT
• Title: Information Geometry of the Geodesic Quantum $f$-Divergences
• Abstract summary: We construct the differential, statistical, and geometrical consequences generated by the geodesic quantum $f$-divergences introduced in [14.<n>For an invertible state $$ and an operator convex function $f$, we compute the Hessian and obtain an explicit formula for the induced monotone quantum information metric $g_,t(f)$.<n>We endow this statistical framework with a geometric meaning in the cone of positive operators.
• Score: 1.9336815376402718
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 $ρ$ and an operator convex function $f$, we compute the Hessian and obtain an explicit formula for the induced monotone quantum information metric $g_{ρ,t}^{(f)}$. 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 $Γ_t$ in the reference purification of a state $σ$ defines a canonical finite binary experiment $(p_t,q_t)$, and introduce a log-likelihood cumulant function $Ψ_{ρ,σ}(t,s)$, 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.

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