Reliability Functions of Quantum Soft Covering and Privacy Amplification via a Mixed-Order Rényi Divergence

In this paper, we introduce a novel mixed-order R\'enyi divergence and investigate its fundamental properties. Using this divergence, we define a family of mixed-order order-two R\'enyi mutual information and R\'enyi conditional entropy. We derive exact reliability functions of quantum soft covering and privacy amplification under the sandwiched R\'enyi divergence with order $\alpha\in[2,\infty)$. The former is jointly characterized by the sandwiched and mixed-order order-two R\'enyi mutual information quantities, while the latter is characterized by the corresponding conditional entropies. These results provide operational interpretations of the proposed mixed-order R\'enyi divergence. To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.