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.