Transpose degradability beyond degradability

Problem

Does there exist a finite-dimensional transpose-degradable quantum channel that is not degradable? Let \(V:A\to B\otimes E\) be an isometry defining a channel and a complementary channel by

\begin{equation} \Phi(X):=\operatorname{Tr}_E(VXV^\dagger), \qquad \Phi^c(X):=\operatorname{Tr}_B(VXV^\dagger). \tag{1} \end{equation}

Equation (1) fixes the output space \(B\) and environment space \(E\). For the transpose \(\mathsf T_E\) in a fixed basis of \(E\), transpose degradability means that a completely positive trace-preserving map \(\mathcal D:\mathcal L(B)\to\mathcal L(E)\) satisfies

\begin{equation} \mathsf T_E\circ\Phi^c=\mathcal D\circ\Phi. \tag{2} \end{equation}

Ordinary degradability instead requires a completely positive trace-preserving map \(\widetilde{\mathcal D}:\mathcal L(B)\to\mathcal L(E)\) satisfying

\begin{equation} \Phi^c=\widetilde{\mathcal D}\circ\Phi. \tag{3} \end{equation}

The question is whether Eq. (2) can hold while no map satisfying Eq. (3) exists.

Source

Singh and Datta explicitly ask whether transpose-degradable channels differ from ordinary degradable channels [SD22]. Brádler had posed the same strict-separation question using the earlier “ conjugate degradable” terminology [Bra15].

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

Writing \(J(\mathcal N)\) for the Choi operator of a channel \(\mathcal N\), Eq. (2) implies

\begin{equation} J(\Phi^c)^{T_E}\geq0, \qquad Q(\Phi)=Q^{(1)}(\Phi) :=\max_{\rho_A} \left[S(\Phi(\rho_A))-S(\Phi^c(\rho_A))\right]. \tag{4} \end{equation}

Thus Eq. (4) gives a PPT complementary Choi operator and additive coherent information, but neither property supplies an ordinary degrading map [SD22].
If \(J(\Phi^c)\) is separable, then \(\Phi^c\) is entanglement breaking and \(\Phi\) is a Hadamard channel, hence degradable. Any strict example must therefore have a PPT-entangled complementary Choi operator. Moreover, transpose-degradable pcubed channels are always ordinarily degradable, so that structured family contains no strict example [Bra15], [SG16].
The universal-cloning family also provides no strict example. For all \(d\geq2\) and \(N,K\geq1\), the optimal symmetric cloner \(\mathcal C_{N\to N+K}^{(d)}\) and optimal pure-state transposition channel \(\mathcal T_{N\to K}^{(d)}\) obey

\begin{equation} \left(\mathcal C_{N\to N+K}^{(d)}\right)^c =\mathcal T_{N\to K}^{(d)}, \qquad \mathcal T_{N\to K}^{(d)}\ \text{is entanglement breaking}, \qquad \mathcal C_{N\to N+K}^{(d)}\ \text{is degradable}. \tag{5} \end{equation}

Equation (5) eliminates the cloning channels that motivated conjugate degradability, but does not prove a general containment theorem [BGS+26].
Reported progress: PR #84 · Issue #105 (withdrawn).

Comment

No strict example and no equality theorem are known. Complementation turns Eq. (2) into transpose antidegradability and Eq. (3) into ordinary antidegradability. Consequently, the source document’s transpose-antidegradable separation question is exactly the same existence problem, not a distinct problem.

References

[SD22]: S. Singh and N. Datta, “ Detecting Positive Quantum Capacities of Quantum Channels,” npj Quantum Information 8, 50 (2022).DOI arXiv
[Bra15]: K. Brádler, “ The Pitfalls of Deciding Whether a Quantum Channel Is (Conjugate) Degradable and How to Avoid Them,” Open Systems & Information Dynamics 22, 1550026 (2015).DOI arXiv
[SG16]: V. Siddhu and R. B. Griffiths, “ Degradable Quantum Channels Using Pure-State to Product-of-Pure-State Isometries,” Physical Review A 94, 052331 (2016).DOI arXiv
[BGS+26]: V. Brzić, D. Grinko, M. Studziński, and M. T. Quintino, “ Optimal Pure State Cloning and Transposition Are Complementary Channels,” arXiv preprint (2026).arXiv

Transpose degradability beyond degradability

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