Converse: Does approximate degradability imply closeness to a degradable channel?

Determine whether every ε-degradable channel Φ, meaning there exists a degrading channel Ξ such that the diamond-norm condition ||Φ^c − Ξ ∘ Φ||_diamond ≤ ε holds, is necessarily within diamond-norm distance θ(ε) of some degradable channel Ψ for a suitable function θ(ε), thereby establishing a converse connection between ε-degradability and ε-close-degradability.

Background

The paper introduces ε-degradable channels, which approximately satisfy the degradability condition, and contrasts this with ε-close-degradable channels, which are close in diamond norm to a truly degradable channel. While the authors prove one direction relating these notions (ε-close-degradable implies (ε+2√ε)-degradable), the reverse implication remains unresolved.

Establishing such a converse would unify the two approximate notions and clarify whether approximate satisfaction of degradability structurally guarantees proximity to a degradable channel, with potential consequences for capacity bounds and channel classification.