Reliability Function of Quantum Information Decoupling via the Sandwiched R\'enyi Divergence

Quantum information decoupling is a fundamental quantum information processing task, which also serves as a crucial tool in a diversity of topics in quantum physics. In this paper, we characterize the reliability function of catalytic quantum information decoupling, that is, the best exponential rate under which perfect decoupling is asymptotically approached. We have obtained the exact formula when the decoupling cost is below a critical value. In the situation of high cost, we provide meaningful upper and lower bounds. This result is then applied to quantum state merging, exploiting its inherent connection to decoupling. In addition, as technical tools, we derive the exact exponents for the smoothing of the conditional min-entropy and max-information, and we prove a novel bound for the convex-split lemma. Our results are given in terms of the sandwiched R\'enyi divergence, providing it with a new type of operational meaning in characterizing how fast the performance of quantum information tasks approaches the perfect.