Exact quantum capacity of the qubit depolarizing channel
For $0 \le p \le 1/3$ let $$\mathcal{D}_p(\rho) = (1 - 3p)\rho + p\,(X\rho X + Y\rho Y + Z\rho Z),$$ where $X, Y, Z$ are the Pauli matrices, so that each Pauli error occurs with probability $p$. By the theorem of Lloyd, Shor and Devetak, the unassisted quantum capacity of $\mathcal{D}_p$ is $$Q(\mathcal{D}_p) = \sup_{k \ge 1} \frac{1}{k} \max_{\rho_k} \Bigl[ S\bigl(\mathcal{D}_p^{\otimes k}(\rho_k)\bigr) - S\bigl((\mathcal{D}_p^c)^{\otimes k}(\rho_k)\bigr) \Bigr],$$ where $\rho_k$ ranges over $k$-qubit density matrices, $S(\rho) = -\operatorname{tr}(\rho \log_2 \rho)$ and $\mathcal{D}_p^c$ is a complementary channel of $\mathcal{D}_p$ from an isometric dilation. Random coding (the hashing bound of Bennett, DiVincenzo, Smolin and Wootters) gives $Q(\mathcal{D}_p) > 0$ for $p$ below about $0.06310$, and DiVincenzo, Shor and Smolin found codes that give $Q(\mathcal{D}_p) > 0$ beyond this point, for $p$ below about $0.06352$. The no-cloning argument of Bruß, DiVincenzo, Ekert, Fuchs, Macchiavello and Smolin gives $Q(\mathcal{D}_p) = 0$ for $p \ge 1/12$. Krohn-Grimberghe proved in exact arithmetic that $Q(\mathcal{D}_p) > 0$ for $p \le 16239/250000 \approx 0.064956$ (2026 preprint).
Problem. Determine $Q(\mathcal{D}_p)$ for every $0 \le p \le 1/3$: an exact formula for the capacity, including the boundary between positive and zero capacity.
Source: Stated by AIM — Open Applied Problems; DiVincenzo, Shor and Smolin (1998), Section VI, ask whether the capacity is positive for noise levels up to the no-cloning threshold.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- IT — Information Theory
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, coherent information, depolarizing channels, quantum channel capacities
References
@article{divincenzo1998,
author = {DiVincenzo, David P. and Shor, Peter W. and Smolin, John A.},
title = {Quantum-channel capacity of very noisy channels},
journal = {Phys. Rev. A},
volume = {57},
number = {2},
year = {1998},
pages = {830--839},
doi = {10.1103/PhysRevA.57.830},
eprint = {quant-ph/9706061},
archivePrefix = {arXiv},
}
@article{lloyd1997,
author = {Lloyd, Seth},
title = {Capacity of the noisy quantum channel},
journal = {Phys. Rev. A},
volume = {55},
number = {3},
year = {1997},
pages = {1613--1622},
doi = {10.1103/PhysRevA.55.1613},
}
@misc{shor2002,
author = {Shor, Peter W.},
title = {The quantum channel capacity and coherent information},
howpublished = {Lecture notes, MSRI Workshop on Quantum Computation},
year = {2002},
}
@article{devetak2005,
author = {Devetak, Igor},
title = {The private classical capacity and quantum capacity of a quantum channel},
journal = {IEEE Trans. Inform. Theory},
volume = {51},
number = {1},
year = {2005},
pages = {44--55},
doi = {10.1109/TIT.2004.839515},
}
@article{bennett1996,
author = {Bennett, Charles H. and DiVincenzo, David P. and Smolin, John A. and Wootters, William K.},
title = {Mixed-state entanglement and quantum error correction},
journal = {Phys. Rev. A},
volume = {54},
number = {5},
year = {1996},
pages = {3824--3851},
doi = {10.1103/PhysRevA.54.3824},
}
@article{bruss1998,
author = {Bru{\ss}, Dagmar and DiVincenzo, David P. and Ekert, Artur and Fuchs, Christopher A. and Macchiavello, Chiara and Smolin, John A.},
title = {Optimal universal and state-dependent quantum cloning},
journal = {Phys. Rev. A},
volume = {57},
number = {4},
year = {1998},
pages = {2368--2378},
doi = {10.1103/PhysRevA.57.2368},
}
@misc{krohngrimberghe2026,
author = {Krohn-Grimberghe, Artus},
title = {A certified lower bound on the quantum-capacity threshold of the depolarizing channel},
year = {2026},
eprint = {2608.15870},
archivePrefix = {arXiv},
}
@misc{aim-depolarizing-quantum-capacity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Exact quantum capacity of the qubit depolarizing channel}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/033-depolarizing-quantum-capacity.md},
note = {Entry 033 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00339, "Exact quantum capacity of the qubit depolarizing channel", clopen.solutions.