Reversible QRT for CQ Channel Conversion

• The paper introduces a novel reversible quantum resource theory framework that overcomes limitations of earlier state-based reductions in CQ channel conversion.
• It employs a generalized quantum Stein's lemma to quantify optimal asymptotic error exponents and establish conversion rates under allowed operations.
• The framework rigorously uses sandwiched Rényi relative entropy to enable full operational reversibility in converting resourceful CQ channels.

A reversible QRT (Quantum Resource Theory) framework for CQ (classical-quantum) channel conversion formalizes the quantitative and operational foundations for interconverting dynamical resources—encoded as CQ channels—under a specified set of allowed operations. The principal advance involves both a precise asymptotic characterization of channel discrimination and the ability to bypass earlier technical limitations in reversible channel conversion, particularly those arising from the state-based reduction and continuity requirements. This article presents the mathematical structure, operational principles, and implications of the reversible QRT framework for CQ channel conversion, with emphasis on the generalized quantum Stein's lemma and its consequences for reversibility of channel interconversion.

1. Foundations of Quantum Resource Theories for Channel Conversion

Quantum resource theories provide an abstract setting in which one distinguishes between "free" and "resourceful" objects, subject to constraints on operations (typically, resource-nongenerating or covariant maps). For static resources (quantum states), the generalized quantum Stein's lemma characterizes the optimal asymptotic error exponent in hypothesis testing between resource and free states by the regularized relative entropy of resource, enabling fully reversible resource interconversion at asymptotic rate determined by this quantity.

When addressing dynamical resources, such as quantum channels or particularly CQ channels (maps from a classical domain $X$ to quantum states: $x \mapsto \rho_x$), analogous resource theories must handle the additional complexity of completely positive trace-preserving maps and the optimization over classical inputs in the discrimination and conversion tasks.

Early extensions to dynamical resources sometimes invoked "state-based techniques" (reducing channels to their Choi states), but this reduction is insufficient for general CQ channel conversion scenarios, especially in channel coding, and required an asymptotic continuity assumption that severely constrained applicability (Hayashi et al., 8 Sep 2025).

2. Generalized Quantum Stein's Lemma for CQ Channel Discrimination

The central technical component is a channel-level generalization of the quantum Stein's lemma—originally formulated for state discrimination—which provides the optimal error exponent for hypothesis testing discrimination between a resource channel $\Phi$ and the set $\mathcal{F}$ of free channels. The appropriate notion of channel divergence is a sandwiched Rényi relative entropy for CQ channels, defined as

$$\widetilde D_\alpha(\Phi_1 \| \Phi_2) := \max_{x \in X} \widetilde D_\alpha\big(\Phi_1(x) \| \Phi_2(x)\big)$$

where $\widetilde D_\alpha(\cdot\|\cdot)$ is the sandwiched Rényi relative entropy between states. This definition captures the "worst-case" discrimination power over all classical inputs.

Key properties of this channel divergence are:

• Additivity: $$\widetilde D_\alpha(\Phi_1^{\otimes n} \| \Phi_2^{\otimes n}) = n \widetilde D_\alpha(\Phi_1 \| \Phi_2)$$ (cf. Lemma “Additivity of channel divergences for CQ channels” (Hayashi et al., 8 Sep 2025)).
• Convergence: $$\widetilde D_\alpha(\Phi_1 \| \Phi_2) \to D(\Phi_1 \| \Phi_2)$$ as $\alpha \to 1$, where $D$ is the Umegaki quantum relative entropy.

The strong converse in channel discrimination is then established by bounding the type-II error exponent in terms of $$\widetilde D_\alpha(\Phi^{\otimes n} \| \mathcal{F})$$, leveraging the worst-case over inputs and bypassing the need for asymptotic continuity (Hayashi et al., 8 Sep 2025).

3. Operational Interpretation and Reversibility in the QRT Framework

The generalized Stein's lemma at the channel level admits a precise operational interpretation: under any allowed (and asymptotically resource-non-generating) set of operations, the asymptotic (many-use) conversion rate between a resource CQ channel $\Phi$ and a target-free channel $\Psi$ is governed by the ratio of their regularized relative entropies of resource. Formally, consider a resource measure $\mathbf{R}(\cdot)$ (typically the regularized relative entropy of resource for channels), then the optimal conversion rate is

$$\text{Rate}_{\mathrm{opt}} = \frac{\mathbf{R}(\Phi)}{\mathbf{R}(\Psi)}$$

provided the set of operations allows for asymptotically resource-non-generating transformations.

This enables a fully reversible interconversion theory analogous to the second law of thermodynamics for static and dynamical resources: any resourceful CQ channel can be asymptotically exchanged for another at a rate fixed by their relative resource content, with the conversion efficiency determined explicitly and operationally via the hypothesis testing error exponents (Hayashi et al., 8 Sep 2025).

4. Technical Resolution of the Multiple-Input Challenge

Unlike the state setting—where a single quantum state suffices—the CQ channel setting requires analysis over all possible classical inputs $x \in X$. This multiple-input challenge is overcome by formulating the sandwiched Rényi channel divergence as a "max-over-inputs" expression:

$$\widetilde D_\alpha(\Phi_1 \| \Phi_2) = \max_{x \in X} \widetilde D_\alpha\big(\Phi_1(x)\|\Phi_2(x)\big)$$

This uniformity is essential: the optimal exponent is dictated by the classical input for which the output states of the two channels are hardest to distinguish, and this directly influences both discrimination and conversion rates. Any protocol that fails to account for this maximum will not achieve the sharp Stein's lemma exponent in the channel setting (Hayashi et al., 8 Sep 2025).

5. Removal of Asymptotic Continuity Constraints

Previous frameworks required an assumption of asymptotic continuity for the resource measure—a technical condition ensuring small perturbations in input do not produce disproportionately large changes in resource quantification. However, this assumption fails to accommodate standard channel coding scenarios, which often involve optimization over highly nonuniform input sequences and non-IID resource structures.

By developing proof techniques directly at the CQ channel level using sandwiched Rényi relative entropy and without reducing to Choi states, the generalized Stein's lemma and corresponding reversible QRT for channel conversion are established without imposing the asymptotic continuity requirement. This broadens operational applicability to core tasks in channel conversion and coding (Hayashi et al., 8 Sep 2025).

6. Applications and Broader Implications

The reversible QRT framework for CQ channel conversion enables:

• General CQ Channel Discrimination: Strong converse error exponents for the hypothesis testing of CQ channels versus arbitrary free sets, with direct applications to quantum channel coding, network communication, and cryptography.
• Unified Conversion Laws: Exact asymptotic conversion rates between arbitrary pairs of CQ channels, subsuming results such as Shannon's channel coding theorem as special cases (when the free class is the set of completely noisy or replacer channels).
• Full Operational Resource Theory: Direct resource quantification for dynamical resources, with additivity and explicit formulas for channel-level regularized resource entropies.

A plausible implication is that the same analytical techniques (sandwiched Rényi relative entropy, max-over-input analysis) can be adapted to further general classes of quantum channels, beyond the CQ setting, as the operational core of a generalized dynamical resource theory.

7. Summary Table: Key Mathematical Elements

Concept	Channel Setting Formulation	Role in Framework
Resource Quantifier	$$\mathbf{R}(\Phi) = \lim_{n\to\infty} \frac{1}{n} D(\Phi^{\otimes n} \| \mathcal{F})$$	Governs conversion rates
Sandwiched Rényi Channel Divergence	$$\widetilde D_\alpha(\Phi_1 \| \Phi_2) = \max_x \widetilde D_\alpha(\Phi_1(x)\|\Phi_2(x))$$	Controls error exponents
Additivity Property	$$\widetilde D_\alpha(\Phi_1^{\otimes n} \| \Phi_2^{\otimes n}) = n \widetilde D_\alpha(\Phi_1 \| \Phi_2)$$	Ensures extensivity of exponents
Reversible Conversion Rate	$\frac{\mathbf{R}(\Phi)}{\mathbf{R}(\Psi)}$	Sets the optimal interconversion
Worst-case Input Maximization	$\max_{x\in X}$	Critical for hypothesis testing

This framework constitutes a complete and operationally meaningful foundation for CQ channel discrimination and reversible conversion, overcoming previous limitations and establishing new quantitative laws for dynamical resources (Hayashi et al., 8 Sep 2025).