Strong Converse Exponents of Quantum Soft Covering and Privacy Amplification

We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders $α\in[\frac{1}{2},\infty)$. For $α\in[\frac{1}{2},1)$, the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for $α\in[1,\infty)$, it is characterized by the order-$α$ sandwiched R{é}nyi mutual information. We also determine the exact strong converse exponent of privacy amplification against quantum side information under the sandwiched R{é}nyi divergence for $α\in(2,\infty)$, expressed in terms of the corresponding order-$α$ sandwiched R{é}nyi conditional entropy. To the best of our knowledge, these results provide the first exact characterization of the strong converse exponent of quantum soft covering and the first precise operational interpretation of the two-parameter club-sandwiched mutual information in the quantum setting. The key ingredient is that we establish the exponential rate of the $K$-functional, which is instrumental in deriving the strong converse exponent of quantum soft covering for $α\in[\frac{1}{2},1)$.