Resource Theories of CQ Channels

• Resource Theories of CQ Channels are frameworks that rigorously quantify the value of channels by defining free operations, resource monotones, and conversion rates.
• They leverage mathematical tools such as relative entropy, hypothesis testing, and sandwiched Rényi divergences to establish operational efficiency and simulation costs.
• This approach enables practical insights into channel discrimination, interconversion, and the optimal use of auxiliary resources in classical-quantum communication.

A resource theory of classical-quantum (CQ) channels is a structured framework for quantifying and analyzing the operational value of CQ channels—maps with classical inputs and quantum outputs—under prescribed free operations and in relation to other resources such as randomness, entanglement, or noiseless communication. This concept generalizes the extensive apparatus of quantum resource theories for states to include dynamical resources, enabling rigorous characterization of channel interconversion, discrimination, and simulation with an explicit focus on operational cost and the role of auxiliary resources.

1. Resource-Theoretic Framework: Free Objects, Operations, and Axioms

A resource theory is predicated on defining the free channels (those accessible at no resource cost) and the allowed class of operations or superchannels (transformations deemed non-resource generating). In the context of CQ channels, the free set is typically chosen to be those channels that output only states with some specific structure, such as classical-quantum (i.e., map a classical label to a quantum state), or in generalized theories, replacer channels (that output a fixed state regardless of input) (Liu et al., 2019, Bergh et al., 16 Sep 2025). The set of free operations is often taken to be all completely positive trace-preserving (CPTP) maps or superchannels that do not generate resourcefulness asymptotically—termed asymptotically resource-non-generating (ARNG) superchannels. These sets satisfy natural axioms:

• Closure under composition and tensor product
• Topological closure (e.g., in the diamond norm)
• Identity channel freeness
• Convexity, permutation-invariance, and existence of a full-rank channel in the free set (Bergh et al., 16 Sep 2025).

In this axiomatic structure, a resource theory of CQ channels elevates channels, rather than just quantum states, as the primary carriers of "resource." This allows one to address operational tasks such as channel discrimination, simulation, and conversion within a coherent framework (Liu et al., 2019, Yuan, 2018, Hayashi et al., 8 Sep 2025).

2. Resource Monotones, Divergences, and Entropic Quantifiers for Channels

Key to any resource theory is the definition of monotones—quantitative functions that do not increase under free operations and accurately track the value of a resource. For CQ channels, a central monotone is the relative entropy of resource, defined as

$$R(\mathcal{E}) = \inf_{F \in \mathcal{F}} \sup_{\nu} D(\mathcal{E}(\nu) \Vert F(\nu))$$

where $D$ is the Umegaki quantum relative entropy and the supremum ranges over all input states $\nu$ (Yuan, 2018, Bergh et al., 16 Sep 2025). The regularized version,

$$R^\infty(\mathcal{E}) = \lim_{n \to \infty} \frac{1}{n} R(\mathcal{E}^{\otimes n})$$

determines the asymptotic interconversion rate in irreversible or reversible settings (Hayashi et al., 5 Aug 2024).

Recent advances exploit the operational link between relative entropy and quantum hypothesis testing, notably via the generalized quantum Stein's lemma (GQSL), which establishes that the regularized channel divergence governs the error exponent in discriminating a resourceful channel from free alternatives (Hayashi et al., 8 Sep 2025, Bergh et al., 16 Sep 2025). Sandwiched Rényi relative entropies,

$$\widetilde{D}_\alpha(\rho\|\sigma) = \frac{1}{\alpha-1} \log \mathrm{Tr}\left[(\sigma^{\frac{1-\alpha}{2\alpha}} \rho \sigma^{\frac{1-\alpha}{2\alpha}})^\alpha\right]$$

with their crucial additivity and monotonicity properties, play a central technical role in establishing sharp strong converse bounds on hypothesis testing rates (Hayashi et al., 5 Aug 2024, Hayashi et al., 8 Sep 2025).

Further monotones such as channel robustness (the minimum noise required to "erase" a resourceful channel into the free set via convex mixing) and conditional min-entropy–based measures permit single-shot and operational characterization outside of the asymptotic regime (Gour, 2016, Liu et al., 2019, Yuan et al., 2020).

3. Simulation, Conversion Rates, and Operational Tasks

The resource theory of CQ channels enables the rigorous paper of simulation and conversion tasks: given a resource channel, how many uses of a standard unit—such as the noiseless identity channel—are required to simulate it (dilution), and conversely, how many instances of a resourceful channel can be distilled from a generic (less resourceful) channel under free operations. For CQ channels, the optimal asymptotic conversion rate $r(\mathcal{E}_1\rightarrow \mathcal{E}_2)$ between two (resourceful) channels $\mathcal{E}_1$ and $\mathcal{E}_2$ is determined by the ratio of their regularized resource monotones: $$r(\mathcal{E}_1\rightarrow \mathcal{E}_2) = \frac{R^\infty(\mathcal{E}_1)}{R^\infty(\mathcal{E}_2)}$$ when both channels are outside the free set (Bergh et al., 16 Sep 2025, Hayashi et al., 8 Sep 2025, Yuan et al., 2020). This statement generalizes the "second law" of resource theories, with $R^\infty$ playing the role of a fundamental monotone analogous to thermodynamic entropy (Hayashi et al., 5 Aug 2024).

For one-shot tasks, single-use and finite-blocklength bounds are controlled by hypothesis testing relative entropy, log-robustness, and (smoothed) channel robustness measures, linking them to operational primitives such as channel simulation error and catalytic dilution/dilation rates (Yuan et al., 2020, Luo, 13 Feb 2025).

4. Hypothesis Testing, Generalized Quantum Stein’s Lemma, and Channel Discrimination

The GQSL for CQ channels formalizes the operational content of resource monotones by linking the error exponent in composite hypothesis testing directly to regularized channel divergences (Hayashi et al., 8 Sep 2025, Bergh et al., 16 Sep 2025). This is achieved under parallel (non-adaptive) discrimination strategies with arbitrary input states, optimizing over all possible input choices (including entangled probes in the general channel case).

The key operational statement is: $$\lim_{n\to\infty} \frac{1}{n} D_H^\epsilon(\mathcal{E}^{\otimes n} \|\mathcal{S}_n) = \lim_{n\to\infty} \frac{1}{n} D(\mathcal{E}^{\otimes n} \|\mathcal{S}_n)$$ where $D_H^\epsilon$ is the hypothesis-testing relative entropy at type-I error $\epsilon$, and $\mathcal{S}_n$ is the allowed alternative (free) set of CQ channels (Bergh et al., 16 Sep 2025). The strict additivity of channel Rényi divergences for CQ (and environment-seizable) channel boxes (Wang et al., 2019, Hayashi et al., 8 Sep 2025) ensures the converse and direct parts of Stein’s lemma hold, consolidating the link between asymptotic error exponents and relative entropy monotones for channel resource discrimination.

5. Trade-offs Among Auxiliary Resources in Channel Simulation

The resource requirements for simulating CQ channels reveal pronounced differences between classical and quantum scenarios. In the classical setting, the reverse Shannon theorem establishes that shared randomness suffices as the sole auxiliary resource, leveling differences in input statistics and enabling efficient simulation using only cbits and rbits (0912.5537). In contrast, simulating general quantum channels (Q-channels) on non-IID inputs may require more sophisticated resources—entanglement-embezzling states, coherence spread, or backward communication—because of entanglement spread among superposed input types, as quantified by the resource-theoretic formalism (0912.5537).

Precise simulation efficiency is captured by resource inequalities and entropic formulas, e.g.,

$$\langle U_F:\rho \rangle \simeq \frac{1}{2} I(R;B)_\rho [q \to q] + \frac{1}{2} I(E;B)_\rho [qq]$$

for quantum channels under coherent feedback, where the forward communication cost and entanglement cost balance according to the structure of the input ensemble and the need to "hide" entanglement spread (0912.5537).

6. Relation to Channel Coding, Capacity, and Communication Scenarios

The resource perspective on CQ channels yields a generalization and refinement of Shannon-theoretic channel capacity, capacity regions, and the ultimate limits of communication when assisted by auxiliary resources. Explicit trade-offs have been derived, e.g., for the simultaneous use of public classical communication, private classical communication, and secret key, yielding polyhedral capacity regions governed by mutual information constraints (Wilde et al., 2010). For certain physically-motivated channels, these regions "single-letterize," admitting capacity-achieving codes and explicit resource allocations.

The general framework is powerful enough to analyze the full spectrum of classical-quantum-communication resources, including strong converse statements for capacity, simulation costs under non-signalling assistance, and operationally meaningful resource monotones (robustness, D_max) linked directly to coding performance (Takagi et al., 2019). These results unify and strengthen the operational foundation for classical-quantum information processing, with broad applicability to practical communication architectures and quantum networks.

7. Mathematical Tools and Technical Developments

The maturity of resource theories for CQ channels is reflected in the advanced mathematical tools deployed:

• Affine resource theories and semidefinite programming yield necessary and sufficient conditions for single-shot conversions, controlled by conditional min-entropy-based monotones (Gour, 2016).
• Generalized robustness, log-robustness, and their asymptotic properties, grounded in convex-split and norm estimates, permit both one-shot and asymptotic analysis of simulation and dilution costs (Liu et al., 2019, Yuan et al., 2020).
• Composite hypothesis testing and permutation invariance arguments, crucial for parallel channel discrimination, extend the machinery of the classical Stein’s lemma and information spectrum method to CQ dynamical resources (Hayashi et al., 8 Sep 2025, Bergh et al., 16 Sep 2025).
• Operational distinctions between worst-case (diamond norm) and average-case (Choi-state) measures ensure that results proved for channel discrimination and simulation under the diamond norm possess maximal robustness and direct operational relevance (Bergh et al., 16 Sep 2025).

Summary Table: Core Quantities in Resource Theories of CQ Channels

Quantity	Definition/Expression	Operational Role
Relative entropy of resource	$$R(\mathcal{E}) = \inf_{F \in \mathrm{Free}} \sup_\nu D(\mathcal{E}(\nu)\Vert F(\nu))$$	Asymptotic conversion rate, resource monotone
Regularized relative entropy	$$R^\infty(\mathcal{E}) = \lim_{n\to\infty} \frac1n R(\mathcal{E}^{\otimes n})$$	Governs reversibility and second law
Channel robustness (log)	$L_R(\mathcal{E}) = \log(1 + R(\mathcal{E}))$	One-shot cost and simulation efficiency
Hypothesis testing relative entropy of channels	$$D_H^\epsilon(\mathcal{E}^{\otimes n}\|\mathcal{S}_n)$$	Strong converse exponent for channel discrimination
Sandwiched Rényi relative entropy for a CQ channel	$$\widetilde{D}_\alpha(\mathcal{E}_1 \Vert \mathcal{E}_2) = \sup_x \widetilde{D}_\alpha(\mathcal{E}_1(x)\|\mathcal{E}_2(x))$$	One-shot/converse bound for testing, rate additivity
Conversion rate between CQ channels	$$r(\mathcal{E}_1 \to \mathcal{E}_2) = R^\infty(\mathcal{E}_1)/R^\infty(\mathcal{E}_2)$$	Asymptotic simulation/distillation/dilution

Significance and Outlook

Resource theories of classical-quantum channels provide a unified operational and mathematical framework for analyzing the value, cost, and interconversion potential of communication and dynamical resources when classical and quantum systems interact. Current results deliver reversible theory (analogous to the thermodynamic second law) anchored in regularized channel divergences and robust against the operational distinctions in the handling of dynamical (as opposed to purely static) scenarios (Hayashi et al., 5 Aug 2024, Hayashi et al., 8 Sep 2025, Bergh et al., 16 Sep 2025). The extension of generalized quantum Stein’s lemma to CQ channels, explicit use of sandwiched Rényi divergences, and the consideration of ARNG operations in the diamond-norm setting collectively provide a comprehensive toolkit for both theoretical investigation and practical protocol design in quantum Shannon theory, channel simulation, and resource management in quantum networks.