Strict inclusion of degradable channels in the less-noisy class

Problem

Do there exist finite-dimensional quantum channels that are less noisy in Watanabe’s regularized sense but are not degradable? Let \(V_{A\to BE}\) be a Stinespring isometry defining a channel and a complementary channel by

\begin{equation} \mathcal N_{A\to B}(X) :=\operatorname{Tr}_E[VXV^\dagger], \qquad \mathcal N^c_{A\to E}(X) :=\operatorname{Tr}_B[VXV^\dagger]. \tag{1} \end{equation}

The channel in Eq. (1) is degradable if there is a completely positive trace-preserving map \(\mathcal D_{B\to E}\) such that

\begin{equation} \mathcal N^c=\mathcal D\circ\mathcal N. \tag{2} \end{equation}

Writing \(P\) for the unassisted private classical capacity, Watanabe calls \(\mathcal N\) less noisy when

\begin{equation} P(\mathcal N^c)=0. \tag{3} \end{equation}

Equivalently, Eq. (3) requires that, for every \(n\ge1\) and every classical–quantum ensemble on \(UA^{n}\), the receiver and environment outputs obey

\begin{equation} I(U:B^n)_{(\operatorname{id}_U\otimes\mathcal N^{\otimes n})(\omega)} \ge I(U:E^n)_{(\operatorname{id}_U\otimes(\mathcal N^c)^{\otimes n})(\omega)}. \tag{4} \end{equation}

Thus the question asks whether the inclusion implied by Eqs. (2)–(4) is strict.

Source

Watanabe proved the inclusion of degradable channels in the regularized less-noisy class and explicitly left open whether it is strict [Wat12]. Hirche and Leditzky later isolated the same strict-inclusion question and related it to superactivation of private capacity [HL23].

Progress

Watanabe proved that every degradable channel satisfies Eq. (3), but left open whether this inclusion is strict [Wat12].
Belzig, Gao, Smith, and Wu constructed flagged mixtures of qubit amplitude-damping channels — including the parameter point \((p,\gamma_1,\gamma_2)=(0.75,0.2,0.81)\) — that are nondegradable yet satisfy the one-copy version of Eq. (4). Their result separates degradability from the level-1 less-noisy condition, while explicitly leaving the all-blocklength condition in Eq. (4) unresolved [BGSW25].
Smith and Wu derived a sufficient condition under which a flagged mixture of degradable and antidegradable channels is regularized less noisy while remaining nondegradable; for their amplitude-damping candidates the condition reduces to strict positivity of a multi-copy comparison coefficient, which they leave unproved, making the construction a conditional route to a separation rather than a counterexample [SW25].
Zhu and Wang subsequently considered the qutrit channel

\begin{equation} \Lambda(X) =\frac12X+\frac14\bigl(\operatorname{Tr}(X)I-X^{\mathsf T}\bigr) \tag{5} \end{equation}

and proved in Theorem 1.1 that

\begin{equation} P(\Lambda)=Q(\Lambda)=0, \qquad \Lambda\ \text{is not antidegradable}. \tag{6} \end{equation}

Setting \(\mathcal N=\Lambda^c\), Eq. (6) gives \(P(\mathcal N^c)=0\), so \(\mathcal N\) is Watanabe-less-noisy. If \(\mathcal N\) were degradable, then \(\Lambda=\mathcal D\circ\Lambda^c\) for some channel \(\mathcal D\), contrary to the non-antidegradability statement in Eq. (6). Hence \(\Lambda^c\) is less noisy but nondegradable [ZW26].

Comment

The answer is affirmative: the complement of the channel in Eq. (5) proves that degradable channels form a proper subset of Watanabe-less-noisy channels. The resolving result [ZW26] is, as of August 2026, a recent preprint; the solved status records its theorem rather than peer-review history. The resolving theorem was checked in version 2 on 6 September 2026.

References

[Wat12]: S. Watanabe, “Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels,” Physical Review A 85, 012326 (2012).DOI arXiv
[BGSW25]: P. Belzig, L. Gao, G. Smith, and P. Wu, “Reverse-Type Data Processing Inequality,” Communications in Mathematical Physics 406, 295 (2025).DOI arXiv
[ZW26]: C. Zhu and X. Wang, “Quantum Incapacity beyond No-Cloning and PPT Mechanisms,” arXiv preprint (2026).arXiv
[HL23]: C. Hirche and F. Leditzky, “Bounding Quantum Capacities via Partial Orders and Complementarity,” IEEE Transactions on Information Theory 69, 283–297 (2023).DOI arXiv
[SW25]: G. Smith and P. Wu, “Additivity of Quantum Capacities in Simple Non-Degradable Quantum Channels,” IEEE Transactions on Information Theory 71, 6134–6154 (2025).DOI arXiv

Strict inclusion of degradable channels in the less-noisy class

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