Regularized less-noisy channels beyond degradability

Problem

Does there exist a finite-dimensional regularized less-noisy quantum channel that is not degradable? Let \(\mathcal T:\mathcal L(\mathcal H_A)\to\mathcal L(\mathcal H_B)\) have a Stinespring isometry \(V:\mathcal H_A\to\mathcal H_B\otimes\mathcal H_E\), defining the channel and one complementary channel by

\begin{equation} \mathcal T(\rho)=\operatorname{Tr}_E(V\rho V^\dagger), \qquad \mathcal T^c(\rho)=\operatorname{Tr}_B(V\rho V^\dagger). \tag{1} \end{equation}

Equation (1) fixes the complementary-channel notation; changing the Stinespring representation only changes \(\mathcal T^c\) by an output isometry.

The channel \(\mathcal T\) is degradable if there exists a channel \(\mathcal D:\mathcal L(\mathcal H_B)\to\mathcal L(\mathcal H_E)\) satisfying

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

Thus Eq. (2) requires exact, rather than approximate, degradability.

For a channel \(\mathcal N:A\to B\) with complement \(\mathcal N^c:A\to E\), let \(\omega_{XBE}:=\sum_x p_x\lvert x\rangle\!\langle x\rvert_X\otimes V_{\mathcal N}\rho_A^xV_{\mathcal N}^\dagger\) for a finite cq ensemble \(\{p_x,\rho_A^x\}\). Define its one-shot private information and private capacity by

\begin{equation} \begin{aligned} \mathcal C_p^{(1)}(\mathcal N) &:=\max_{\{p_x,\rho_A^x\}} \bigl[I(X;B)_\omega-I(X;E)_\omega\bigr],\\ \mathcal C_p(\mathcal N) &:=\sup_{n\ge1}\frac1n \mathcal C_p^{(1)}(\mathcal N^{\otimes n}). \end{aligned} \tag{3} \end{equation}

The regularization in Eq. (3) distinguishes the question from its single-letter counterpart. In particular, define

\begin{equation} \mathfrak D :=\{\mathcal T:\mathcal T\text{ satisfies Eq.~(2)}\}. \tag{4} \end{equation}

\begin{equation} \mathfrak L_1 :=\{\mathcal T:\mathcal C_p^{(1)}(\mathcal T^c)=0\}. \tag{5} \end{equation}

\begin{equation} \mathfrak L_\infty :=\{\mathcal T:\mathcal C_p(\mathcal T^c)=0\}. \tag{6} \end{equation}

\begin{equation} \mathfrak D\subseteq\mathfrak L_\infty\subseteq\mathfrak L_1. \tag{7} \end{equation}

A channel in \(\mathfrak L_\infty\), as defined in Eq. (6), is regularized less noisy. The problem is whether the first inclusion in Eq. (7) is strict.

Source

Belzig, Gao, Smith, and Wu explicitly state that it remains open whether a regularized less-noisy channel can fail to be degradable [BGSW25]. Hirche and Leditzky previously isolated the same strict-inclusion question and related it to superactivation of private capacity [HL23].

Progress

Data processing proves \(\mathfrak D\subseteq\mathfrak L_\infty\). Watanabe showed that regularized less-noisy channels have single-letter quantum and private capacities, \(\mathcal Q(\mathcal T)=\mathcal C_p(\mathcal T) =\mathcal Q^{(1)}(\mathcal T)\) [Wat12]. Hirche and Leditzky further proved that superactivation of private capacity would imply \(\mathfrak D\subsetneq\mathfrak L_\infty\); neither result decides whether the inclusion is strict [HL23].
Belzig et al. constructed explicit flagged mixtures of qubit amplitude-damping channels that belong to \(\mathfrak L_1\setminus\mathfrak D\), including the parameter point \((p,\gamma_1,\gamma_2)=(0.75,0.2,0.81)\). This proves \(\mathfrak D\subsetneq\mathfrak L_1\), but the construction’s membership in \(\mathfrak L_\infty\) is unknown and therefore does not settle Eq. (7) [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; they leave that positivity unproved. Their construction is therefore a conditional route to a separation, not a counterexample [SW25].
Zhu and Wang’s Theorem 1.1 gives a finite-dimensional channel \(\Lambda\) with \(\mathcal C_p(\Lambda)=0\) that is not antidegradable [ZW26]. Taking \(\mathcal T=\Lambda^c\) yields \(\mathcal C_p(\mathcal T^c)=0\). Degradability of \(\mathcal T\) would make \(\Lambda\) antidegradable, so \(\mathcal T\in\mathfrak L_\infty\setminus\mathfrak D\).

Comment

The existence question is answered affirmatively by the complementary-channel argument in Progress [ZW26]. The cited result is an arXiv preprint (version 2, checked 6 September 2026); this status does not assert peer review. This is the same archived question as “Strict inclusion of degradable channels in the less-noisy class.” Both permanent identities are retained.

References

[Wat12]: S. Watanabe, “Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels,” Physical Review A 85, 012326 (2012).DOI 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
[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

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