Quantum $f$-divergences and Their Local Behaviour: An Analysis via Relative Expansion Coefficients

2025-10-07

AI Summary

This study addresses the problem of information loss and preservation in quantum state distinguishability under quantum channels, aiming to unify the characterization of global (via $f$-divergences) and local (via Riemannian seminorms) relative expansion behaviors. Methodologically, it integrates the data processing inequality, local second-order expansions, monotone Riemannian seminorms, $chi^2$-divergence analysis, and channel primitivity techniques. The main contributions are: (i) the introduction of a “relative expansion coefficient” framework; (ii) the first construction of a novel family of $f$-functions ensuring strict equality between global and local coefficients; (iii) an equivalence theory enabling qualitative property transfer across coefficients and revealing their intrinsic connection to quantum channel reversibility; and (iv) theoretical results showing that positive expansion coefficients quantitatively lower-bound the convergence rate of quantum Markov processes and yield a reverse convergence theorem for approximate information recovery.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Content-based information diffusion
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Any reasonable measure of distinguishability of quantum states must satisfy a data processing inequality, that is, it must not increase under the action of a quantum channel. We can ask about the proportion of information lost or preserved and this leads us to study contraction and expansion coefficients respectively, which can be combined into a single emph{relative expansion coefficient} for study. We focus on two prominent families: (i) standard quantum $f$ divergences and (ii) their local (second-order) behaviour, which induces a monotone Riemannian semi-norm (that is linked to the $chi^2$ divergence). Building on prior work, we identify new families of $f$ for which the global ($f$ divergence) and local (Riemannian) relative expansion coefficients coincide for every pair of channels, and we clarify how exceptional such exact coincidences are. Beyond equality, we introduce an emph{equivalence} framework that transfers qualitative properties such as strict positivity uniformly across different relative expansion coefficients. Leveraging the link between equality in the data processing inequality (DPI) and channel reversibility, we apply our framework of relative expansion coefficients to approximate recoverability of quantum information. Using our relative expansion results for primitive channels, we prove a reverse quantum Markov convergence theorem, converting positive expansion coefficients into quantitative lower bounds on the convergence rate.

Problem

• Analyzing contraction and expansion coefficients for quantum state distinguishability measures
• Establishing equivalence between global and local quantum f-divergence expansion coefficients
• Applying expansion coefficients to quantum information recovery and Markov convergence

Innovation

• Developed relative expansion coefficients framework
• Identified coinciding global and local expansion coefficients
• Applied expansion coefficients to quantum recovery