Reliability Function of Quantum Information Decoupling

Quantum information decoupling is an important quantum information processing task, which has found broad applications. In this paper, we characterize the performance of catalytic quantum information decoupling regarding the exponential rate under which perfect decoupling is approached, namely, the reliability function. We have obtained the exact formula when the decoupling cost is not larger than a critical value. In the situation of high cost, we provide upper and lower bounds. This result is then applied to quantum state merging and correlation erasure, exploiting their connection to decoupling. As technical tools, we derive the exponents for smoothing 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 operational meanings in characterizing how fast the performance of quantum information tasks approach the perfect.