Non-Linear Strong Data-Processing for Quantum Hockey-Stick Divergences

2025-12-18

AI Summary

Existing linear strong data processing inequalities (SDPIs) for the quantum hockey-stick divergence suffer from insufficient tightness, limiting precision in quantifying information loss under quantum channels. Method: We establish the first *nonlinear* SDPI in the quantum setting by introducing a generalized Dobrushin-type $F_gamma$ curve to characterize contraction behavior under heterogeneous channel concatenations, and derive an inverse Pinsker-type $f$-divergence inequality constrained by the hockey-stick divergence. Our approach integrates quantum information theory, $f$-divergence analysis, nonlinear inequality construction, and quantum differential privacy modeling. Contribution/Results: The proposed nonlinear SDPI significantly improves bound tightness over linear counterparts, yielding sharper finite mixing-time upper bounds for quantum Markov chains. Moreover, it provides enhanced quantum local differential privacy guarantees for sequential private quantum channels—enabling more accurate and robust characterization of distinguishability decay in both classical and quantum regimes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Data-processing is a desired property of classical and quantum divergences and information measures. In information theory, the contraction coefficient measures how much the distinguishability of quantum states decreases when they are transmitted through a quantum channel, establishing linear strong data-processing inequalities (SDPI). However, these linear SDPI are not always tight and can be improved in most of the cases. In this work, we establish non-linear SDPI for quantum hockey-stick divergence for noisy channels that satisfy a certain noise criterion. We also note that our results improve upon existing linear SDPI for quantum hockey-stick divergences and also non-linear SDPI for classical hockey-stick divergence. We define $F_γ$ curves generalizing Dobrushin curves for the quantum setting while characterizing SDPI for the sequential composition of heterogeneous channels. In addition, we derive reverse-Pinsker type inequalities for $f$-divergences with additional constraints on hockey-stick divergences. We show that these non-linear SDPI can establish tighter finite mixing times that cannot be achieved through linear SDPI. Furthermore, we find applications of these in establishing stronger privacy guarantees for the composition of sequential private quantum channels when privacy is quantified by quantum local differential privacy.

Problem

• Establishes non-linear strong data-processing inequalities for quantum hockey-stick divergences.
• Improves existing linear and classical non-linear bounds for noisy quantum channels.
• Applies results to enhance privacy guarantees in sequential private quantum channels.

Innovation

• Develop non-linear strong data-processing inequalities for quantum hockey-stick divergences
• Generalize Dobrushin curves to quantum setting for sequential channel composition
• Derive reverse-Pinsker inequalities with constraints on hockey-stick divergences