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

TL;DR

The work develops non-linear strong data-processing inequalities for the quantum hockey-stick divergence, proving that certain noisy channels satisfy a non-linear contraction bound that tightens classical linear SDPI. It introduces F_\gamma curves to analyze channel composition, establishes mixing-time bounds that can be finite under non-linear contraction, and connects these results to stronger privacy guarantees under sequential private quantum channels via quantum local differential privacy. In addition, the authors derive reverse Pinsker-type inequalities relating hockey-stick divergences to a broad class of f-divergences and provide practical criteria for containment in B^{\gamma,\delta} and implications for privacy composition. Overall, the results yield tighter information-processing guarantees and have direct applications to quantum privacy, channel mixing, and the analysis of sequential quantum operations.

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.

Figures (2)

• Figure 2: Comparison of Upper Bounds on Mixing Times with Linear and Non-Linear SDPI: We consider $\mathcal{N} \in \mathcal{B}^{\gamma, \delta}$ with $\gamma=8$ and $\delta=0$. With that, we plot the upper bounds on $t_{\gamma'}^{\mathcal{N}}(\beta)$ given in \ref{['prop:mixing_times1']} (marked as Non-linear) and \ref{['prop:mixing_linear_gam_del']} (marked as Linear) for $\gamma'=3$.
• Figure 3: Comparing LDP bounds on the relative entropy. Dashed lines represent Equation \ref{['Eq:dasgupta']} (in particular a lower bound on that since we chose $\lambda=m$) and solid lines our new bound in Equation \ref{['Eq:RE-LDP-bound']}. Left: Plot over $\lambda$, respectively $m$, for fixed $\epsilon=\{1,2,3\}, \delta=0.01, \tau=0.25$. Right: Plot over $\epsilon$ for fixed $\delta=\{0.1,0.2,0.3\}, \lambda=m=0.1, \tau=0.25$.

Theorems & Definitions (68)

• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 1
• proof
• Corollary 2
• proof
• Proposition 3
• proof