Remove asymptotic continuity by a worst-case input and diamond-norm formulation

Develop a generalization of Theorem 2 (the second law for classical-quantum channels) that replaces the Choi-state-based regularized relative entropy of resource with the regularized channel divergence defined via worst-case input states (\tilde{R}_R in equation (eq:channel_divergence)) and uses the diamond norm for the convergence distance (in place of trace distance on Choi states in equation (eq:conversion_rate_methods)), thereby eliminating the asymptotic continuity requirement (equation (eq:condition_asymptotic_continuity)) on the operations.

Background

In the channel setting, the authors base their second-law formulation on Choi states to avoid non-IID worst-case inputs and to ensure tractable asymptotics. This approach requires an explicit asymptotic continuity condition on the operations, because the corresponding linear maps on Choi operators need not be trace-non-increasing.

A formulation based on channel divergences with worst-case inputs and diamond-norm distances would be more conventional for channels and could remove the asymptotic continuity assumption, but it would require overcoming additional non-IID challenges beyond those addressed in their current proof techniques.

References

It was conjectured in that a similar AEP holds for channels when smoothing is done with respect to the diamond norm.

— The regularized channel Rényi divergence is continuous  (2609.29885 - Schmitt et al., 24 Sep 2026) in Section 4.3, “Channel asymptotic equipartition property”

Along with these limitations, as explained above, it is not straightforward to show another generalization of Theorem~\ref{thm:second_law} using worst-case inputs to the channels (i.e., using $\tilde{R}_\mathrm{R}$ in~eq:channel_divergence and the diamond norm for the distance in~eq:conversion_rate_methods) to eliminate the requirement of the asymptotic continuity~eq:condition_asymptotic_continuity, which we leave for future work.

— Generalized Quantum Stein's Lemma and Second Law of Quantum Resource Theories  (2408.02722 - Hayashi et al., 2024) in Methods, The second law of QRTs for states and classical-quantum (CQ) channels