Quantum Information Ordering and Differential Privacy

We study quantum differential privacy (QDP) by defining a notion of the order of informativeness between pairs of quantum states. In particular, we show that if the hypothesis testing divergence of one pair dominates over that of the other pair, then this dominance holds for every $f$-divergence. This approach completely characterizes $(\varepsilon,\delta)$-QDP mechanisms by identifying the most informative $(\varepsilon,\delta)$-DP quantum state pairs. We apply this to 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.