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.

翻译：量子信息解耦是一项基础的量子信息处理任务，同时也是量子物理学众多领域中的关键工具。本文刻画了催化量子信息解耦的可靠性函数，即渐近逼近完美解耦的最佳指数速率。当解耦成本低于临界值时，我们得到了精确公式。在高成本情形下，我们给出了有意义的上界和下界。随后，利用其与解耦的内在联系，我们将此结果应用于量子态合并。此外，作为技术工具，我们推导了条件最小熵与最大信息平滑化的精确指数，并证明了凸分裂引理的一个新颖界。我们的结果以夹层Rényi散度给出，这为其在刻画量子信息任务性能趋近完美的速率方面提供了一种新型的操作意义。