All-code exponential strong converse for degradable channels

Problem

Does every finite-dimensional degradable quantum channel satisfy an all-code exponential strong converse for quantum communication at its quantum capacity? Let \(V:A\to B\otimes E\) be a Stinespring isometry, and define the channel and one complementary channel by

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

The channel in Eq. (1) is degradable when there is a completely positive trace-preserving map \(\mathcal D:\mathcal L(B)\to\mathcal L(E)\) such that

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

For a channel satisfying Eq. (2), its quantum capacity is the single-letter coherent information

\begin{equation} Q(\mathcal N)=Q^{(1)}(\mathcal N) :=\max_{\rho_A} \left[S(\mathcal N(\rho_A))-S(\mathcal N^c(\rho_A))\right], \qquad S(\tau):=-\operatorname{Tr}(\tau\log_2\tau). \tag{3} \end{equation}

For an arbitrary \(n\)-use entanglement-transmission code, let the encoder and decoder be CPTP maps \(\mathcal E_n:\mathcal L(S_n)\to\mathcal L(A^{\otimes n})\) and \(\mathcal R_n:\mathcal L(B^{\otimes n})\to\mathcal L(\widehat S_n)\), with \(\dim S_n=\dim\widehat S_n=M_n\). If \(\Phi_{M_n}^{R_nS_n}\) is maximally entangled, define the code rate and entanglement fidelity by

\begin{equation} r_n:=\frac{1}{n}\log_2 M_n, \qquad F_n:=\operatorname{Tr}\!\left[ \Phi_{M_n}^{R_n\widehat S_n} \bigl(\operatorname{id}_{R_n}\otimes \mathcal R_n\circ\mathcal N^{\otimes n}\circ\mathcal E_n\bigr) (\Phi_{M_n}^{R_nS_n}) \right]. \tag{4} \end{equation}

The all-code exponential strong converse asks whether, for every \(R>Q(\mathcal N)\), there are constants \(\gamma_R>0\) and \(n_R\) such that every code in Eq. (4) obeys

\begin{equation} r_n\geq R,\qquad n\geq n_R \quad\Longrightarrow\quad F_n\leq 2^{-\gamma_R n}. \tag{5} \end{equation}

Equation (5) requires a bound for every encoder–decoder pair, rather than for almost every member of a random code ensemble.

Source

Morgan and Winter explicitly isolated the upgrade from their pretty-strong converse to a full strong converse for degradable channels [MW14]. Wilde records this gap, together with the corresponding all-code gap for the quantum erasure channel, in Section 24.10 of his text [Wil17].

Progress

Morgan and Winter proved that rates above Eq. (3) force the asymptotic fidelity to be bounded away from one for every degradable channel. Their “pretty strong” converse did not force \(F_n\) to zero and therefore did not imply Eq. (5) [MW14].
Wilde and Winter proved exponential fidelity decay above the quantum capacity of the erasure channel for almost all codes drawn from their random ensemble. The exceptional set was not excluded, so their theorem was not the all-code statement in Eq. (5) [WW14].
Kondra, Brinster, Kampermann, Bruß, and Wyderka prove Eq. (5) for every finite-dimensional degradable channel and also prove an exponential strong converse for every finite-dimensional antidegradable channel. Since every quantum erasure channel is degradable or antidegradable, their result gives the first all-code exponential strong converse for the erasure channel over its full parameter range [KBK+26].

Comment

The solved status records the theorem claimed in [KBK+26]. As of 2026-09-02, that work was a one-month-old, unrefereed v1 preprint, so the closure is provisional and should be re-audited if the preprint changes. Problem asks for an exponential strong converse for the larger, potentially strict class of transpose-degradable channels; the theorem above applies to ordinarily degradable and antidegradable channels and does not by itself settle that extension.

References

[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
[WW14]: M. M. Wilde and A. Winter, “Strong Converse for the Quantum Capacity of the Erasure Channel for Almost All Codes,” in 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014), LIPIcs 27, 52–66 (2014).DOI arXiv
[Wil17]: M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press (2017), Sec. 24.10.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

All-code exponential strong converse for degradable channels

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