Quantum Information Ordering and Differential Privacy
This work addresses foundational challenges in quantum privacy for researchers in quantum information theory and differential privacy, providing theoretical tools and bounds that generalize classical results to quantum settings.
The paper tackles the problem of characterizing quantum differential privacy (QDP) by defining an ordering of informativeness between quantum state pairs, showing that dominance in hypothesis testing divergence implies dominance for all f-divergences, and applies this to analyze stability in quantum learning algorithms and derive tight bounds for privatized hypothesis testing and parameter estimation.
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.