Geodesic Quantum $f$-Divergences

• URL: http://arxiv.org/abs/2608.15833v1
• Date: Sun, 16 Aug 2026 16:07:56 GMT
• Title: Geodesic Quantum $f$-Divergences
• Abstract summary: We show strengthened geodesic quantum quantum-divergences $D_ft$, $0leq tleq1.<n>They reduce to the $(,z)$ Petz and Rényi divergences joining the Petz and Umegaki families.<n>We also determine parameter-monity regimes and the intersections with the $(,z)$ family.
• Score: 1.9336815376402718
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce the geodesic quantum $f$-divergences $D_f^t$, $0\leq t\leq1$, obtained from the affine-invariant geodesic between the standard and maximal relative modular operators. They reduce to the classical $f$-divergence for commuting states. The logarithmic generator yields geodesic relative entropies joining the Umegaki and Belavkin-Staszewski entropies, while the power generators yield $(t,α)$-Rényi divergences joining the Petz and geometric families. Our first main result is data processing of $D_f^t$ for every finite operator-convex generator $f$ and every $t\in[0,1]$. In particular, this gives DPI for the geodesic relative entropies and for the $(t,α)$-Rényi divergences when $0<α<1$ or $1<α\leq2$. Our second main result identifies equality in the DPI: for invertible states, every equality-determining operator-convex generator has, at each nonmaximal parameter $0\leq t<1$, exactly the Petz sufficiency class; at $t=1$, this changes to the generally larger maximal, or BS, class, which also coincides with the equality class of the divergences associated with the quadratic generator for every $t$. Our third main result is the corresponding collapse of invertible geodesic quantum Markov chains: for each of the three ordered conditional-mutual-information constructions, the $t$-quantum Markov chains are precisely the quantum Markov chains for $0\leq t<1$, whereas at $t=1$ they are the invertible BS quantum Markov chains, a class that can be strictly larger. We also determine parameter-monotonicity regimes and the intersections with the $(α,z)$ family. We prove strengthened data-processing and reconstruction estimates; we compare the three conditional orientations; we establish continuity bounds under positive lower-eigenvalue assumptions together with complementary discontinuity results; and we give a capacity-per-unit-cost interpretation.

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