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 Renyi divergence, providing it with operational meanings in characterizing how fast the performance of quantum information tasks approach the perfect.
In this paper, we introduce a novel mixed-order Rényi divergence and investigate its fundamental properties. Using this divergence, we define a family of mixed-order order-two Rényi mutual information and Rényi conditional entropy. We derive exact reliability functions of quantum soft covering and privacy amplification under the sandwiched Rényi divergence with order
α∈[2,∞). The former is jointly characterized by the sandwiched and mixed-order order-two Rényi mutual information quantities, while the latter is characterized by the corresponding conditional entropies. These results provide operational interpretations of the proposed mixed-order Rényi divergence. To the best of our knowledge, this is the first exact characterization of the reliability function for quantum soft covering.
We determine the exact strong converse exponent of quantum soft covering under the sandwiched R{é}nyi divergence for all orders
α∈[21,∞). For
α∈[21,1), the exponent is characterized by the two-parameter club-sandwiched mutual information, whereas for
α∈[1,∞), 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∈(2,∞), 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∈[21,1).