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Universal strong converse for the classical capacity

Solved ID op_55869a5fec880498 Last edited 8 September 2026
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Does every finite-dimensional quantum channel satisfy the strong converse property at its unassisted classical capacity, or does some channel admit a sequence of codes above capacity whose average success probability stays bounded away from zero?

Fix finite-dimensional Hilbert spaces \(A\) and \(B\) and a quantum channel (completely positive trace-preserving map) \(\mathcal N:\mathcal L(A)\to\mathcal L(B)\). With the von Neumann entropy \(S(\tau):=-\operatorname{Tr}(\tau\log_2\tau)\), define the Holevo capacity

\begin{equation} \chi(\mathcal N):=\max_{\{p_x,\rho_x\}} \Bigl[S\Bigl(\sum_x p_x\,\mathcal N(\rho_x)\Bigr) -\sum_x p_x\,S\bigl(\mathcal N(\rho_x)\bigr)\Bigr], \tag{1} \end{equation}

where the maximum runs over all finite ensembles \(\{p_x,\rho_x\}\) of states \(\rho_x\in\mathcal D(A)\), and define the classical capacity

\begin{equation} C(\mathcal N):=\lim_{n\to\infty}\frac1n\,\chi\bigl(\mathcal N^{\otimes n}\bigr), \tag{2} \end{equation}

which exists by superadditivity of Eq. (1) and Fekete’s lemma.

For \(\varepsilon\in(0,1)\) and blocklength \(n\), let \(M^*(n,\varepsilon)\) be the largest \(M\) for which there are codeword states \(\rho_m\in\mathcal D(A^{\otimes n})\), \(m=1,\dots,M\), arbitrary across the \(n\) uses (inputs entangled across channel uses are allowed), and a POVM \(\{D_m\}_{m=1}^M\) on \(B^{\otimes n}\) with average error at most \(\varepsilon\),

\begin{equation} \frac1M\sum_{m=1}^M \Bigl(1-\operatorname{Tr}\bigl[D_m\,\mathcal N^{\otimes n}(\rho_m)\bigr]\Bigr) \le\varepsilon. \tag{3} \end{equation}

The channel \(\mathcal N\) has the strong converse property if

\begin{equation} \limsup_{n\to\infty}\frac1n\,\log_2 M^*(n,\varepsilon)\le C(\mathcal N) \tag{4} \end{equation}

holds for every fixed \(\varepsilon\in(0,1)\); equivalently, the \(\varepsilon\)-capacity is independent of \(\varepsilon\), and every rate \(R>C(\mathcal N)\) forces the optimal average success probability to vanish as \(n\to\infty\). Achievability, \(\liminf_{n\to\infty}\frac1n\log_2 M^*(n,\varepsilon)\ge C(\mathcal N)\) for every \(\varepsilon\in(0,1)\), is the Holevo–Schumacher–Westmoreland theorem, so the question is whether Eq. (4) holds for every finite-dimensional \(\mathcal N\). A negative answer consists of an explicit channel \(\mathcal N\), constants \(R>C(\mathcal N)\) and \(\varepsilon\in(0,1)\), and an infinite sequence of blocklengths with codes of size at least \(2^{nR}\) and average error at most \(\varepsilon\) in the sense of Eq. (3). The resource model is unassisted, memoryless use of \(\mathcal N^{\otimes n}\), without entanglement assistance or feedback: the strong converse for the entanglement-assisted classical capacity already holds for every channel, so the unassisted codes of Eq. (3) are the case archived here, and the regularization in Eq. (2) means the question is not reducible to additivity of Eq. (1).

Source

König and Wehner pose the question explicitly: their introduction asks for “the validity of the strong converse property for a general quantum channel” and they settle unital qubit, depolarizing, and Werner–Holevo channels [KW09]. Wilde, Winter, and Yang prove the strong converse for all entanglement-breaking and Hadamard channels and state further strong-converse theorems as an open question in their conclusion [WWY14].

Progress

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

  • Hayashi and Nagaoka gave a general capacity formula and an information-spectrum characterization of the strong converse property for classical–quantum channels [HN03]. Extending that characterization to fully quantum inputs, where the codeword states in Eq. (3) may be entangled across channel uses, is a natural route and is not carried out there.

  • König and Wehner proved Eq. (4) for all unital qubit channels, the \(d\)-dimensional depolarizing channel, and the Werner–Holevo channel, by relating the strong converse to additivity of minimum output entropies; their converse allows code states entangled across channel uses, matching Eq. (3) [KW09].

  • Wilde, Winter, and Yang extended Eq. (4) to all entanglement-breaking channels and all Hadamard channels, the complementary channels of the former, by bounding the success probability with a sandwiched Rényi relative entropy whose channel version is subadditive on these classes; strong converses previously known for particular covariant channels emerge as special cases [WWY14].

  • The classical capacity of the quantum erasure channel obeys Eq. (4), obtained by Wilde and Winter as a side result of their almost-all-codes analysis of the erasure channel’s quantum capacity [WW14]. For phase-insensitive optical channels under a photon-number occupation constraint on the codebook (the paper explicitly notes the strong converse need not hold under the usual mean-photon-number constraint), Bardhan, García-Patrón, Wilde, and Winter proved the strong converse for the classical capacity [BGP+15], an infinite-dimensional counterpart outside the finite-dimensional scope posed above.

  • In the assisted setting the strong converse is universal: Gupta and Wilde proved it for the entanglement-assisted classical capacity of every finite-dimensional channel, from multiplicativity of completely bounded \(p\)-norms [GW15]. Boche, Janßen, and Kaltenstadler showed that the strong converse nevertheless fails for the entanglement-assisted classical capacity of compound quantum channels, so any affirmative answer must use the fixed memoryless model of Eq. (3) rather than a compound or arbitrarily varying one [BJK17].

  • Earlier results for general channels established strong-converse bounds above the capacity in Eq. (2): Wang, Xie, and Duan derived two semidefinite-programming bounds such that any code whose rate exceeds the bound has success probability vanishing exponentially [WXD18], and Wang, Fang, and Tomamichel showed that the regularization of the \(\Upsilon\)-information upper-bounds the classical capacity and that for covariant channels the \(\Upsilon\)-information is a strong converse bound [WFT19]. Neither establishes Eq. (4) at \(C(\mathcal N)\) for every channel.

  • Other recent work targets adjacent quantities rather than the unassisted transmission strong converse: Ye, Bergh, and Datta prove strong-converse bounds on the classical identification capacity of the qubit depolarizing channel, using the regularized Holevo capacity of Eq. (2) as an established quantity [YBD26]; Singh proves that the entanglement-assisted transmission capacity is a strong-converse bound for identification [Sin26]; and the Lean-QIT formalization machine-checks the entanglement-assisted strong converse but no unassisted counterpart [ZTZ+26]. These results concern different operational tasks.

  • Cheng and Tomamichel prove the universal strong converse for the classical capacity of every finite-dimensional quantum channel in Theorem 5, Eq. (25), with the proof in Appendix E of arXiv:2609.08998v1 [CT26]. The result allows arbitrary inputs entangled across memoryless channel uses and arbitrary decoding POVMs, exactly as in Eq. (3). Their Theorem 4 gives continuity of the regularized sandwiched Rényi capacity at order one; combined with Theorem 5, it yields a strictly positive strong-converse exponent at every fixed rate \(R>C(\mathcal N)\). Consequently the optimal average success probability decays exponentially, establishing Eq. (4) in the full archived scope.

Comment

Solved by the universal classical-capacity strong converse of Cheng and Tomamichel [CT26], Theorem 5 and Appendix E. The resolving source is the September 2026 arXiv preprint, version 1; this status records that complete claimed theorem and does not claim peer-reviewed publication. Its scope is the fixed, finite-dimensional, unassisted memoryless channel in the statement, including codewords entangled across uses. It does not assert a strong converse for compound channels or infinite-dimensional channels under arbitrary energy constraints. The historical question is retained here as a solved archive.

References

[HN03]
M. Hayashi and H. Nagaoka, “General Formulas for Capacity of Classical-Quantum Channels,” IEEE Transactions on Information Theory 49(7), 1753–1768 (2003).arXiv
[KW09]
R. König and S. Wehner, “A Strong Converse for Classical Channel Coding Using Entangled Inputs,” Physical Review Letters 103, 070504 (2009).DOIarXiv
[WWY14]
M. M. Wilde, A. Winter, and D. Yang, “Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy,” Communications in Mathematical Physics 331, 593–622 (2014).DOIarXiv
[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).DOIarXiv
[BGP+15]
B. R. Bardhan, R. García-Patrón, M. M. Wilde, and A. Winter, “Strong Converse for the Classical Capacity of Optical Quantum Communication Channels,” IEEE Transactions on Information Theory 61(4), 1842–1850 (2015).DOIarXiv
[GW15]
M. K. Gupta and M. M. Wilde, “Multiplicativity of Completely Bounded \(p\)-Norms Implies a Strong Converse for Entanglement-Assisted Capacity,” Communications in Mathematical Physics 334, 867–887 (2015).DOIarXiv
[BJK17]
H. Boche, G. Janßen, and S. Kaltenstadler, “Entanglement-Assisted Classical Capacities of Compound and Arbitrarily Varying Quantum Channels,” Quantum Information Processing 16, 88 (2017).DOIarXiv
[WXD18]
X. Wang, W. Xie, and R. Duan, “Semidefinite Programming Strong Converse Bounds for Classical Capacity,” IEEE Transactions on Information Theory 64(1), 640–653 (2018).DOIarXiv
[WFT19]
X. Wang, K. Fang, and M. Tomamichel, “On Converse Bounds for Classical Communication over Quantum Channels,” IEEE Transactions on Information Theory 65(7), 4609–4619 (2019).DOIarXiv
[YBD26]
L. Ye, B. Bergh, and N. Datta, “Strong Converse Bounds on the Classical Identification Capacity of the Qubit Depolarizing Channel,” arXiv preprint (2026).arXiv
[Sin26]
S. Singh, “The Entanglement-Assisted Transmission Capacity Is a Strong Converse Bound for Identification,” arXiv preprint (2026).arXiv
[ZTZ+26]
C. Zhu, Z. Tang, G. Zhen, Y. Cao, Y. Zhao, R. Chen, X. Zhao, L. Zhang, and X. Wang, “Lean-QIT: Towards a Formal Infrastructure for Quantum Information Theory,” arXiv preprint (2026).arXiv
[CT26]
H.-C. Cheng and M. Tomamichel, “No information transmission through quantum channels above capacity,” arXiv preprint, version 1, 8 September 2026.arXiv

Cite this problem

Please also cite the primary sources listed under References. Cite this page for the statement, status, and stable identifier.

BibTeX

@incollection{qiqcop_op_55869a5fec880498,
  title = {Universal strong converse for the classical capacity},
  booktitle = {Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo)},
  year = {2026},
  howpublished = {\url{https://qiqc-op.com/problem/op_55869a5fec880498/}},
  note = {Stable ID op_55869a5fec880498; status: Solved; accessed 2026-09-27}
}

Plain text

“Universal strong converse for the classical capacity,” Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo), https://qiqc-op.com/problem/op_55869a5fec880498/, ID op_55869a5fec880498, accessed 2026-09-27.

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op_55869a5fec880498
01M207X5AKR9Z7V4T07VAX027M