  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.
