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 Rényi entropy. This can be viewed as a parametric generalization of König 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 Rényi 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 Rényi information and sandwiched Rényi capacity, respectively.