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.

翻译：我们通过定义量子态对之间的信息量级概念来研究量子差分隐私（QDP）。具体而言，我们证明若一对量子态的假设检验散度支配另一对量子态的散度，则该支配关系对所有$f$-散度均成立。此方法通过识别最具信息量的$(\varepsilon,\delta)$-差分隐私量子态对，完整刻画了$(\varepsilon,\delta)$-QDP机制。我们将其应用于分析量子差分隐私学习算法的稳定性，将经典结果推广至$\delta>0$的情形。此外，我们研究了私有化假设检验与私有化量子参数估计的精确极限，包括QDP约束下量子费舍尔信息的紧致上界。最后，我们针对曲棍球散度建立了差分隐私量子通道的近似最优收缩界。