Generalized Quantum Stein's Lemma for Classical-Quantum Dynamical Resources

Abstract

Channel conversion constitutes a pivotal paradigm in information theory and its applications to quantum physics, providing a unified problem setting that encompasses celebrated results such as Shannon's noisy-channel coding theorem. Quantum resource theories (QRTs) offer a general framework to study such problems under a prescribed class of operations, such as those for encoding and decoding. In QRTs, quantum states serve as static resources, while quantum channels give rise to dynamical resources. A recent major advance in QRTs is the generalized quantum Stein's lemma, which characterizes the optimal error exponent in hypothesis testing to discriminate resource states from free states, enabling a reversible QRT framework for static resources where asymptotic conversion rates are fully determined by the regularized relative entropy of resource. However, applications of QRTs to channel conversion require a framework for dynamical resources. The earlier extension of the reversible framework to a fundamental class of dynamical resources, represented by classical-quantum (CQ) channels, relied on state-based techniques and imposed an asymptotic continuity assumption on operations, which prevented its applicability to conventional channel coding scenarios. To overcome this problem, we formulate and prove a generalized quantum Stein's lemma directly for CQ channels, by developing CQ-channel counterparts of the core proof techniques used in the state setting. Building on this result, we construct a reversible QRT framework for CQ channel conversion that does not require the asymptotic continuity assumption, and show that this framework applies to the analysis of channel coding scenarios. These results establish a fully general toolkit for CQ channel discrimination and conversion, enabling their broad application to core conversion problems for this fundamental class of channels.