Generalized Quantum Stein's Lemma for Classical-Quantum Dynamical Resources (2509.07271v1)

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.

• The paper extends Quantum Stein's lemma to CQ channels, transitioning the hypothesis testing framework from static quantum states to dynamic resources.
• It introduces key operational metrics, like the diamond norm and channel divergence, to quantify error exponents and analyze channel conversion.
• The study establishes a reversible resource theory for CQ channel conversion, deriving exact conversion rates and capacity bounds without continuity assumptions.

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

Introduction and Motivation

The paper establishes a generalized quantum Stein's lemma for classical-quantum (CQ) channels, extending the resource-theoretic hypothesis testing framework from quantum states (static resources) to CQ channels (dynamical resources). This generalization is motivated by the need to analyze channel conversion problems in quantum information theory, where CQ channels represent a tractable yet nontrivial class of dynamical resources. The work addresses limitations in previous approaches that relied on state-based techniques and imposed asymptotic continuity assumptions, which are incompatible with operationally relevant scenarios such as channel coding.

CQ Channels and Resource Theory Framework

CQ channels are defined as maps $\Phi: \mathcal{X} \to \mathcal{D}(\mathcal{H})$, where $\mathcal{X}$ is a finite set of classical inputs and $\mathcal{D}(\mathcal{H})$ is the set of density operators on a finite-dimensional Hilbert space. The paper introduces operationally meaningful metrics for CQ channels, notably the diamond norm, which captures worst-case input distinguishability, and the channel divergence, which generalizes quantum relative entropy to channels.

A resource theory for CQ channels is formulated by specifying a set of free operations (superchannels) and a family of free channels $\mathcal{F}$, satisfying axioms of full-rankness, compactness, closure under tensor product, and convexity. The regularized relative entropy of resource, defined via channel divergence, serves as the central monotone for asymptotic resource conversion.

Generalized Quantum Stein's Lemma for CQ Channels

The main technical result is a direct proof of the generalized quantum Stein's lemma for CQ channels. The hypothesis testing task is to distinguish $n$ IID copies of a resourceful CQ channel from a non-IID set of free CQ channels, optimizing over classical inputs and output measurements. The optimal error exponent for the type II error is shown to be given by the regularized channel divergence between the resource channel and the set of free channels:

$$\lim_{n \to \infty} -\frac{1}{n} \log \beta_\epsilon(\Phi^{\otimes n} \| \mathcal{F}) = \lim_{n \to \infty} \frac{1}{n} D(\Phi^{\otimes n} \| \mathcal{F})$$

where $\beta_\epsilon$ is the minimal type II error under a type I error constraint $\epsilon$, and $D$ is the channel divergence.

The proof adapts key techniques from the state setting—pinching, information spectrum methods, and error-exponent bounds based on sandwiched Rényi divergences—to the CQ channel context. The classical nature of the inputs is crucial for extending these methods, as the analysis of fully quantum (QQ) channels remains intractable.

Reversible Resource Theory for CQ Channel Conversion

Building on the generalized Stein's lemma, the paper constructs a reversible resource theory for CQ channel conversion. The conversion rate between two CQ channels under asymptotically resource-non-generating superchannels is exactly characterized by the ratio of their regularized relative entropies of resource:

$$r_{\tilde{\mathcal{O}}}(\Phi_{\mathrm{in}} \to \Phi_{\mathrm{out}}) = \frac{R_{\mathrm{R}}^\infty(\Phi_{\mathrm{in}})}{R_{\mathrm{R}}^\infty(\Phi_{\mathrm{out}})}$$

where $R_{\mathrm{R}}^\infty$ is the regularized relative entropy of resource. This result holds without the asymptotic continuity assumption required in previous work, making the framework applicable to conventional channel coding scenarios.

The paper further provides a characterization of $R_{\mathrm{R}}^\infty$ in terms of the logarithmic generalized robustness, establishing tight connections between operational and resource-theoretic quantities.

Applications to Channel Capacity and Conversion

The framework is applied to derive bounds on channel capacities and conversion rates for CQ channels. When the set of free channels consists of replacer channels (which have zero capacity), the regularized relative entropy of resource coincides with the channel capacity. The conversion rate between CQ channels under various classes of superchannels (deterministic, classically correlated, entanglement-assisted, non-signaling) is shown to be bounded above by the ratio of their capacities, with equality in several operationally relevant scenarios.

The analysis recovers known results for channel coding and reverse Shannon theorems, and extends them to settings with entanglement or randomness assistance, without requiring additional continuity assumptions.

Technical Contributions and Numerical Results

• Strong numerical result: The optimal asymptotic conversion rate between CQ channels is exactly given by the regularized relative entropy of resource, even under the strictly harder diamond-norm approximation and more relaxed class of operations.
• Contradictory claim to previous work: The framework does not require asymptotic continuity, in contrast to earlier approaches, and is strictly more general.
• Additivity and subadditivity: The paper proves additivity of channel divergence and subadditivity of resource measures for CQ channels, enabling the existence of regularized quantities.
• Minimax characterization: The type II error in hypothesis testing admits a minimax representation, facilitating analysis and optimization.

Implications and Future Directions

The results establish a unified and reversible resource-theoretic framework for CQ channel discrimination and conversion, with direct operational relevance to quantum communication and coding. The techniques developed provide a general toolkit for analyzing dynamical resources, bridging the gap between static and dynamical resource theories.

The extension to fully quantum (QQ) channels remains an open problem, as the lack of classical inputs precludes the use of current proof techniques. Future work may focus on developing new analytical tools for QQ channels, exploring connections to quantum combs, and investigating resource theories for more general classes of operations.

Conclusion

The paper rigorously generalizes the quantum Stein's lemma to CQ channels, constructs a reversible resource theory for channel conversion, and demonstrates its applicability to channel capacity analysis. The framework dispenses with restrictive continuity assumptions, enabling broad application to core problems in quantum information theory. The results clarify the role of CQ channels as a fundamental class of dynamical resources and provide a robust foundation for further developments in quantum resource theories and operational quantum information.