Operational Interpretation of the Sandwiched Rényi Divergence of Order 1/2 to 1 as Strong Converse Exponents

Abstract

We provide the sandwiched Rényi divergence of order $\alpha \in (\frac{1}{2},1)$, as well as its induced quantum information quantities, with an operational interpretation in the characterization of the exact strong converse exponents of quantum tasks. Specifically, we consider (a) smoothing of the max-relative entropy, (b) quantum privacy amplification, and (c) quantum information decoupling. We solve the problem of determining the exact strong converse exponents for these three tasks, with the performance being measured by the fidelity or purified distance. The results are given in terms of the sandwiched Rényi divergence of order $\alpha \in (\frac{1}{2},1)$, and its induced quantum Rényi conditional entropy and quantum Rényi mutual information. This is the first time to find the precise operational meaning for the sandwiched Rényi divergence with Rényi parameter in the interval $\alpha \in (\frac{1}{2},1)$.

V2: minor changes. V3: more discussions and clarifications added, small errors and incompleteness in proof fixed