Quantum Information Ordering and Differential Privacy

TL;DR

The paper develops a quantum generalization of Blackwell-style informativeness, introducing a quantum hypothesis-testing based order to compare DP mechanisms. It proves that dominance in quantum hypothesis-testing divergences implies dominance across all quantum $f$-divergences, enabling a complete characterization of $(\varepsilon,\delta)$-DP quantum mechanisms via the weakest DP pair $(\rho_{(\varepsilon,\delta)},\sigma_{(\varepsilon,\delta)})$. This framework yields precise limits for privatized hypothesis testing and quantum parameter estimation (via the maximal SLD Fisher information), and it establishes near-optimal contraction bounds for $(\varepsilon,\delta)$-DP channels with respect to hockey-stick divergences. The results extend classical DP stability to the quantum regime ($\delta>0$) and connect privacy to stability (Holevo information) and channel contraction, providing a unified, operational view of privacy-information trade-offs in quantum learning and inference.

Abstract

We study quantum differential privacy (QDP) by defining a notion of the order of informativeness between two pairs of quantum states. In particular, we show that if the hypothesis testing divergence of the one pair dominates over that of the other pair, then this dominance holds for every $f$-divergence. This approach completely characterizes $(\varepsilon,δ)$-QDP mechanisms by identifying the most informative $(\varepsilon,δ)$-DP quantum state pairs. We apply this to analyze the stability of quantum differentially private learning algorithms, generalizing classical results to the case $δ>0$. Additionally, we study precise limits for privatized hypothesis testing and privatized quantum parameter estimation, including tight upper-bounds on the quantum Fisher information under QDP. Finally, we establish near-optimal contraction bounds for differentially private quantum channels with respect to the hockey-stick divergence.

Figures (3)

• Figure 1: Graphical representation of $\mathcal{R}(\varepsilon,\delta)$ : characteristic region of $(\varepsilon,\delta)$-QDP (in shaded region), where we define the boundaries (a), (b), (c) and (d) to be $\beta = 1- \delta - e^\varepsilon \alpha$, $\beta = e^{-\varepsilon} (1-\delta-\alpha)$, $\beta = 1 - e^{-\varepsilon}(\alpha - \delta)$ and $\beta = e^\varepsilon(1-\alpha) + \delta$ respectively and $\mathcal{R}(\varepsilon,\delta)$ has two fixed points $(\frac{1-\delta}{1+e^\varepsilon},\frac{1-\delta}{1+e^\varepsilon})$ and $(\frac{e^\varepsilon + \delta}{1+e^\varepsilon}, \frac{e^\varepsilon + \delta}{1+e^\varepsilon})$ as its extremal points, where the former and the latter points are known to be the worst and the best fixed points respectively from the perspective of privacy.
• Figure 2: Graphical representation of $\mathcal{R}(\rho'_{(\varepsilon,\delta)},\sigma'_{(\varepsilon,\delta)} )$ (in the whole shaded region) which is strictly larger than $\mathcal{R}(\varepsilon,\delta)$ (in inner shaded region), where the two extremal points of $\mathcal{R}(\rho'_{(\varepsilon,\delta)},\sigma'_{(\varepsilon,\delta)} )$ outside $\mathcal{R}(\varepsilon,\delta)$ are $\left(\frac{1 -\delta}{1+e^\varepsilon} - \delta,\frac{1 -\delta}{1+e^\varepsilon} - \delta\right)$ and $\left(\frac{e^\varepsilon+\delta}{1 + e^\varepsilon} +\delta, \frac{e^\varepsilon+\delta}{1 + e^\varepsilon} +\delta\right)$.
• Figure 3: Privacy based learning framework.

Theorems & Definitions (92)

• Definition 1
• Definition 2: Frequency type classes
• Definition 3: RGBS21
• Definition 4: Classical minimun type-II error
• Definition 5: Classical hypothesis testing divergence
• Definition 6: Classical hockey-stick divergenceHull2003
• Definition 7: Classical alpha divergence
• Definition 8
• Definition 9: Quantum minimun type-II error
• Definition 10: Quantum hypothesis testing divergence Wang_2012