Transpose-antidegradability and antidegradability

Determine whether every transpose-antidegradable quantum channel whose complement is completely less noisy than the channel itself must be antidegradable.

Background

Transpose antidegradability provides a regularized comparison and guarantees zero private capacity, but it does not generally provide the reference-complete comparison needed to rule out activation with antidegradable helpers. The paper shows that failure of complete less-noisy ordering yields a finite-dimensional half-erasure activator, whereas satisfaction of that ordering rules out every antidegradable helper. The authors state that the remaining implication is unresolved, and note that the activating examples do not settle it because both fail the complete order.

References

By \cref{prop:erasure} and tensorization of the complete comparison, this is equivalent to the unresolved implication \begin{equation} \left.\begin{gathered} \cN\text{ transpose-antidegradable},\ \cNc\succeq_{\rm ln}{\mathrm c}\cN \end{gathered}\right} \stackrel{?}{\Longrightarrow}\ \cN\text{ antidegradable}. \end{equation}

— Quantum estimation, channel orders, and private capacity  (2610.03706 - Hirche, 2 Oct 2026) in Section 6, Discussion; displayed implication immediately following the paragraph beginning “Whether complete unsmoothed measured-$\chi^2$ ordering…”