Reliability Functions of Quantum Soft Covering and Privacy Amplification via a Mixed-Order R\'enyi Divergence

In this paper, we introduce a novel mixed-order Rényi divergence and investigate its fundamental properties.

Using this divergence, we define a family of mixed-order order-two Rényi mutual information and Rényi conditional entropy.

We derive exact reliability functions of quantum soft covering and privacy amplification under the sandwiched Rényi divergence with order $\alpha\in[2,\infty)$.

The former is jointly characterized by the sandwiched and mixed-order order-two Rényi mutual information quantities, while the latter is characterized by the corresponding conditional entropies.

These results provide operational interpretations of the proposed mixed-order Rényi divergence.

To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.