Maximal α-Leakage for Quantum Privacy Mechanisms

In this work, maximal α-leakage is introduced to quantify how much a quantum adversary can learn about any sensitive information of data upon observing its disturbed version via a quantum privacy mechanism. We first show that an adversary's maximal expected α-gain using optimal measurement is characterized by measured conditional R\'enyi entropy. This can be viewed as a parametric generalization of K\"onig et al.'s famous guessing probability formula [IEEE Trans. Inf. Theory, 55(9), 2009]. Then, we prove that the α-leakage and maximal α-leakage for a quantum privacy mechanism are determined by measured Arimoto information and measured R\'enyi capacity, respectively. Various properties of maximal α-leakage, such as data processing inequality and composition property are established as well. Moreover, we show that regularized α-leakage and regularized maximal α-leakage for identical and independent quantum privacy mechanisms coincide with α-tilted sandwiched R\'enyi information and sandwiched R\'enyi capacity, respectively.