Operational Interpretation of the Sandwiched R\'enyi Divergence of Order 1/2 to 1 as Strong Converse Exponents

TL;DR

This paper establishes the operational meaning of the sandwiched Rényi divergence for orders between 1/2 and 1, linking it to strong converse exponents in quantum information tasks like privacy amplification and decoupling.

Contribution

It provides the first precise operational interpretation of the sandwiched Rényi divergence for the parameter range (1/2, 1), connecting it to key quantum information exponents.

Findings

Determined the exact strong converse exponents for quantum tasks.

Linked sandwiched Rényi divergence to quantum privacy amplification and decoupling.

Provided new operational meanings for quantum Rényi quantities.

Abstract

We provide the sandwiched R\'enyi 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 R\'enyi divergence of order $\alpha\in(\frac{1}{2},1)$, and its induced quantum R\'enyi conditional entropy and quantum R\'enyi mutual information. This is the first time to find the precise operational meaning for the sandwiched R\'enyi divergence with R\'enyi parameter in the interval $\alpha\in(\frac{1}{2},1)$.