Can entangled codewords improve amplitude-damping classical capacity?
For $0 < \gamma < 1$ let $\mathcal{A}_\gamma$ be the qubit amplitude damping channel $$\mathcal{A}_\gamma(\rho) = K_0 \rho K_0^\dagger + K_1 \rho K_1^\dagger, \qquad K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix}.$$ For a finite-dimensional quantum channel $\Phi$ let $$\chi(\Phi) = \sup_{\{p_j, \rho_j\}} \Bigl[ S\Bigl( \sum_j p_j \Phi(\rho_j) \Bigr) - \sum_j p_j S\bigl( \Phi(\rho_j) \bigr) \Bigr]$$ be its Holevo information, the supremum taken over finite ensembles of input density operators, with $S(\rho) = -\operatorname{Tr}(\rho \log_2 \rho)$. The unassisted classical capacity of $\mathcal{A}_\gamma$ is $\sup_{n \ge 1} \chi(\mathcal{A}_\gamma^{\otimes n})/n$, where the input ensembles may be entangled across the $n$ uses. Product ensembles give $\chi(\mathcal{A}_\gamma^{\otimes n}) \ge n \chi(\mathcal{A}_\gamma)$, and Hastings showed that the Holevo information is not additive in general.
Problem. Is $\chi(\mathcal{A}_\gamma^{\otimes n}) = n \chi(\mathcal{A}_\gamma)$ for every $n \ge 1$ and every $0 < \gamma < 1$? Equivalently, does the unassisted classical capacity $\sup_{n \ge 1} \chi(\mathcal{A}_\gamma^{\otimes n})/n$ equal the one-use Holevo information $\chi(\mathcal{A}_\gamma)$?
Source: Leditzky, Kaur, Datta and Wilde (2018), Introduction, for the classical capacity of the amplitude damping channel. Stated by AIM — Open Applied Problems as additivity of the Holevo information.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- IT
- Keywords
- AIM Open Applied Problems, amplitude damping channel, classical capacity, Holevo information
References
@article{leditzky2018,
author = {Leditzky, Felix and Kaur, Eneet and Datta, Nilanjana and Wilde, Mark M.},
title = {Approaches for approximate additivity of the {H}olevo information of quantum channels},
journal = {Phys. Rev. A},
volume = {97},
number = {1},
year = {2018},
pages = {012332},
doi = {10.1103/PhysRevA.97.012332},
eprint = {1709.01111},
archivePrefix = {arXiv},
}
@article{hastings2009,
author = {Hastings, M. B.},
title = {Superadditivity of communication capacity using entangled inputs},
journal = {Nature Phys.},
volume = {5},
number = {4},
year = {2009},
pages = {255--257},
doi = {10.1038/nphys1224},
}
@misc{aim-amplitude-damping-additivity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Can entangled codewords improve amplitude-damping classical capacity?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/207-amplitude-damping-additivity.md},
note = {Entry 207 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We determine the unassisted classical capacity of every qubit generalized amplitude-damping channel, for all damping strengths and thermal occupations. It equals the one-shot Holevo capacity and is given by a one-variable maximum. A binary pure-state ensemble attains the one-use optimum, and its independent products attain the optimum at every block length. We also prove that one-shot Holevo capacity, minimum output …
Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with those of any finite-dimensional partner channel. This class includes all qubit-to-qubit channels of Kraus …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00661, "Can entangled codewords improve amplitude-damping classical capacity?", clopen.solutions.