Strong converse for quantum capacity of degradable channels

Prove the strong converse for the quantum capacity of all degradable channels by establishing equality Q(N) = \widehat{Q}(N), where Q(N) is the weak converse quantum capacity and \widehat{Q}(N) is the strong converse quantum capacity.

Background

The strong converse property for quantum capacity asserts that any rate exceeding the capacity Q(N) inevitably leads to error tending to one; it is known to hold for certain channel classes (e.g., PPT entanglement-binding and generalized dephasing channels).

For degradable channels, only a "pretty strong" converse is established, and the full strong converse remains conjectured. Settling this conjecture would unify the capacity behavior for degradable channels and directly impact bounds used throughout the paper, including those connecting resolvability and identification capacities.

References

For general degradable channels it is conjectured, but only a “pretty strong” converse is known, meaning that the asymptotic rate of quantum codes is bounded by Q(\cN) for sufficiently small error .

— Quantum soft-covering lemma with applications to rate-distortion coding, resolvability and identification via quantum channels  (2306.12416 - Atif et al., 2023) in Section 5 (Quantum channel resolvability), Remark