Quantum Soft Covering and Decoupling with Relative Entropy Criterion
Xingyi He, Touheed Anwar Atif, S. Sandeep Pradhan
TL;DR
This work develops quantum soft covering and decoupling results under a relative entropy criterion, covering both fully quantum and classical-quantum (CQ) channels. It provides tight 1-shot and asymptotic bounds by connecting covering rates to coherent information and Holevo information, respectively, and introduces new continuity and quadratic bounds for relative entropy to enable these results. The authors extend the quantum decoupling theorem to the relative-entropy setting, obtaining tighter guarantees than existing trace-norm-based results via Pinsker’s inequality. Overall, the relative-entropy framework yields sharper covering and decoupling performance, with implications for channel resolvability and secure quantum information processing.
Abstract
We propose quantum soft covering problems for fully quantum channels and classical-quantum (CQ) channels using relative entropy as a criterion of operator closeness. We prove covering lemmas by deriving one-shot bounds on the rates in terms of smooth min-entropies and smooth max-divergences, respectively. In the asymptotic regime, we show that for quantum channels, the rate infimum defined as the logarithm of the minimum rank of the input state is the coherent information between the reference and output state; for CQ channels, the rate infimum defined as the logarithm of the minimum number of input codewords is the Helovo information between the input and output state. Furthermore, we present a one-shot quantum decoupling theorem with relative entropy criterion. Our results based on the relative-entropy criterion are tighter than the corresponding results based on the trace norm considered in the literature due to the Pinsker inequality.