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

TL;DR

This paper introduces non-linear strong data-processing inequalities for quantum hockey-stick divergences, improving bounds over existing linear inequalities and enhancing privacy guarantees in quantum information processing.

Contribution

It establishes non-linear SDPI for quantum hockey-stick divergence, generalizes Dobrushin curves to quantum channels, and applies results to quantum privacy guarantees.

Findings

Non-linear SDPI provide tighter bounds than linear SDPI.

Results improve privacy guarantees for sequential quantum channels.

Derived reverse-Pinsker inequalities for constrained f-divergences.

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 $F_\gamma$ 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.