Quantum Information Ordering and Differential Privacy

TL;DR

This paper introduces a framework for quantum differential privacy based on the informativeness order of quantum states, providing new bounds and characterizations for privacy mechanisms in quantum information processing.

Contribution

It defines a new notion of informativeness order for quantum states and characterizes quantum differential privacy mechanisms using divergence dominance, with applications to hypothesis testing and quantum estimation.

Findings

Dominance in hypothesis testing divergence implies dominance in all f-divergences.

Characterization of privacy mechanisms via most informative quantum state pairs.

Derived tight bounds on quantum Fisher information under quantum differential privacy.

Abstract

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.