Geodesic Quantum $f$-Divergences

2026-08-16

AI Summary

This study addresses the absence of a unified geometric framework and data processing inequality (DPI) for quantum f-divergences by proposing a novel family of geodesic quantum f-divergences constructed via affine-invariant geodesics. This formulation unifies classical definitions, including Umegaki and Belavkin-Staszewski (BS) entropies. We rigorously establish the DPI and its equality conditions, characterize quantum Markov chain structures, and reveal an expansion phenomenon within the BS class at t=1. Furthermore, continuity bounds and a unit-cost capacity interpretation are derived. Collectively, this work provides quantum information theory with new tools that integrate geometric intuition with rigorous analytical properties, thereby resolving fundamental theoretical gaps in the field.

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.

Problem

• Geodesic quantum f-divergences
• Data processing inequality
• Equality conditions
• Quantum Markov chains

Innovation

• Geodesic Quantum f-Divergences
• Data Processing Inequality
• Belavkin-Staszewski Entropy
• Quantum Markov Chains
• Operator Convex Generator