Exponential strong converse for transpose-degradable channels

Problem

Does every finite-dimensional transpose-degradable channel satisfy an exponential strong converse for quantum communication at its single-letter quantum capacity? Let \(V:A\to B\otimes E\) be an isometry and suppose that

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

where \(\mathsf T_E\) is transpose in a fixed basis of \(E\) and \(\mathcal G:\mathcal L(B)\to\mathcal L(E)\) is completely positive and trace preserving. For the channel in Eq. (1), transpose degradability gives

\begin{equation} Q(\Phi)=Q^{(1)}(\Phi) :=\max_{\rho_A} \left[S(\Phi(\rho_A))-S(\Phi^c(\rho_A))\right], \qquad S(\sigma):=-\operatorname{Tr}(\sigma\log_2\sigma). \tag{2} \end{equation}

Equation (2) fixes the rate threshold.

At blocklength \(n\), let \(\mathcal E_n:\mathcal L(S_n)\to\mathcal L(A^{\otimes n})\) and \(\mathcal R_n:\mathcal L(B^{\otimes n})\to\mathcal L(\widehat S_n)\) be arbitrary encoder and decoder channels, with \(\dim R_n=\dim S_n=\dim\widehat S_n=M_n\). Define the maximally entangled target by

\begin{equation} \varphi_{M_n}:= |\varphi_{M_n}\rangle\!\langle\varphi_{M_n}|, \qquad |\varphi_{M_n}\rangle :=\frac1{\sqrt{M_n}}\sum_{i=1}^{M_n}|i\rangle_{R_n}|i\rangle_{S_n}. \tag{3} \end{equation}

Using the target in Eq. (3), let

\begin{equation} \omega_n:= \left(\operatorname{id}_{R_n}\otimes \mathcal R_n\circ\Phi^{\otimes n}\circ\mathcal E_n\right)(\varphi_{M_n}), \qquad r_n:=\frac1n\log_2M_n, \qquad F_n:=\operatorname{Tr}(\varphi_{M_n}^{R_n\widehat S_n}\omega_n). \tag{4} \end{equation}

The problem is whether, for every channel in Eq. (1), the quantities in Eq. (4) satisfy

\begin{equation} \forall R>Q(\Phi)\ \exists\,\gamma_R>0,\ n_R\in\mathbb N: \quad r_n\geq R,\ n\geq n_R \ \Longrightarrow\ F_n\leq2^{-\gamma_R n} \tag{5} \end{equation}

for every encoder–decoder sequence. Equation (5) is the all-code exponential strong-converse property at the threshold in Eq. (2).

Source

Singh and Datta supply the complex-linear transpose-degradable formulation and its single-letter capacity [SD22]. Morgan and Winter explicitly note that extending their degradable-channel converse method to the corresponding conjugate-degradable class requires arguments not provided there; together these papers pose the present class-wide question implicitly [MW14].

Progress

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

Transpose degradability proves the tensor-power identity

\begin{equation} Q^{(1)}(\Phi^{\otimes n})=nQ^{(1)}(\Phi) \qquad(n\geq1). \tag{6} \end{equation}

Equation (6) establishes the capacity formula in Eq. (2), but gives no finite-block upper bound on \(F_n\) [SD22].
For every ordinarily degradable channel \(\mathcal N\) and fixed purified-distance error \(\varepsilon<1/\sqrt2\), Morgan and Winter proved

\begin{equation} \log_2 N_E(n,\varepsilon\mid\mathcal N) \leq nQ^{(1)}(\mathcal N)+O(\sqrt{n\log n}), \tag{7} \end{equation}

where \(N_E\) is the largest entanglement-generation code dimension. Equation (7) is a pretty-strong converse for ordinary degradability, and their proof does not extend to transpose-degradable channels solely from coherent-information additivity [MW14].
Kondra et al. proved in 2026 that every finite-dimensional ordinarily degradable or antidegradable channel \(\mathcal N\) obeys, for every rate above capacity,

\begin{equation} r_n\geq R>Q(\mathcal N) \quad\Longrightarrow\quad F_n\leq2^{-\gamma_R n} \quad\text{for all sufficiently large }n. \tag{8} \end{equation}

Equation (8) settles the known transpose-degradable examples that are also degradable, but its theorem does not cover a hypothetical strict transpose-degradable channel [KBK+26].
Beigi and Tomamichel’s Theorem 1 proves an exponential strong converse for every finite-dimensional memoryless quantum channel, including arbitrary encoders and joint decoders. Applying it to \(\Phi\) with the rate gap \(R-Q(\Phi)>0\) proves Eq. (5); Eq. (2) identifies the threshold with the single-letter capacity. This argument covers transpose-degradable channels whether or not they are ordinarily degradable [BT26].
Reported progress: PR #38.

Comment

Updated on 2026-09-15. The complete resolution follows from Theorem 1 of the September 2026 version-1 preprint [BT26], also recorded in the general quantum-capacity strong-converse problem. This preprint is not identified as peer-reviewed at this audit; the Solved status records its complete theorem with that publication caveat. The existence of a strictly transpose-degradable channel remains a separate question: the converse theorem neither constructs such a channel nor proves that none exists.

References

[SD22]: S. Singh and N. Datta, “ Detecting Positive Quantum Capacities of Quantum Channels,” npj Quantum Information 8, 50 (2022).DOI arXiv
[MW14]: C. Morgan and A. Winter, “ Pretty Strong Converse for the Quantum Capacity of Degradable Channels,” IEEE Transactions on Information Theory 60, 317–333 (2014).DOI arXiv
[KBK+26]: T. V. Kondra, R. Brinster, H. Kampermann, D. Bruß, and N. Wyderka, “ Sharp Quantum Capacity Thresholds: Exponential Strong Converses for Degradable and Antidegradable Channels,” arXiv preprint (2026).arXiv
[BT26]: S. Beigi and M. Tomamichel, “Strong Converse for Quantum Capacity via a Fully Quantum Blowing-Up Lemma,” arXiv preprint, version 1, 10 September 2026.arXiv

Exponential strong converse for transpose-degradable channels

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