Quantum capacity of the depolarizing and qubit Pauli channels

The depolarizing channel leaves a qubit untouched with probability $1-3p$ and applies each of the Pauli errors $X$ , $Y$ and $Z$ with probability $p$ . How many qubits per use can it carry reliably? Even for this simplest noise model the answer is unknown, and so is the noise level at which it drops to zero.

Problem

A qubit Pauli channel with error probabilities $\mathbf p=(p_I,p_X,p_Y,p_Z)$ acts on a qubit density matrix $\rho$ as

\Lambda_{\mathbf p}(\rho)=p_I\,\rho+p_X\,X\rho X+p_Y\,Y\rho Y+p_Z\,Z\rho Z .

Its quantum capacity $Q(\Lambda_{\mathbf p})$ is the highest rate, in qubits per channel use, at which quantum states can be sent with vanishing error by block codes over many independent uses, with no shared entanglement or classical side channel. By the quantum channel coding theorem it is the regularized coherent information [3]

Q(\Lambda)=\lim_{n\to\infty}\frac1n\max_{\rho_n}\Big[S\big(\Lambda^{\otimes n}(\rho_n)\big)-S\big((\Lambda^{\otimes n})^{c}(\rho_n)\big)\Big],

where $S$ is the von Neumann entropy, $\Lambda^{c}$ is the complementary channel to the environment, and $\rho_n$ ranges over states of $n$ input qubits.

Question. Determine $Q(\Lambda_{\mathbf p})$ for every $\mathbf p$ , including the exact boundary of the region where it is positive. The central case is the depolarizing channel $\mathbf p=(1-3p,p,p,p)$ : find $Q$ as a function of $p$ , or at least the threshold $p^{*}$ above which $Q=0$ .

Background

In 1996 Bennett, DiVincenzo, Smolin and Wootters showed that random codes achieve the hashing rate $1-H(\mathbf p)$ , where $H$ is the Shannon entropy of the error distribution [1]; it equals the coherent information of a single use with a maximally mixed input. In 1998 DiVincenzo, Shor and Smolin found codes that beat the hashing rate for the very noisy depolarizing channel [2], so the answer needs many-use inputs. These codes gain through degeneracy, where distinct errors act identically on the encoded information, and the depolarizing channel has remained the prototypical test case for this effect [5].

What is known

Exact values: the dephasing channel ( $p_X=p_Y=0$ ) is degradable, so its capacity is the single-letter coherent information [4].
Depolarizing channel, lower side, in the convention above: the hashing rate is positive up to $p=0.063096$ [5]; concatenated repetition codes reach $p=0.06352$ [2, 5]; Fern and Whaley (2008) reached $p=0.06376$ [5]; and Agarwal et al. (2026) reach $p=0.064657$ , found numerically with rank-two inputs on the symmetric subspace of 45 qubits [5].
Krohn-Grimberghe (2026) certified in exact rational arithmetic a 45-copy rank-two input with positive coherent information at $p=16239/250000=0.064956$ , in an unrefereed preprint whose certificate comes with an independent verifier. Since this coherent information can only decrease as $p$ grows, $Q>0$ for all $p\le0.064956$ [6].
Upper side: a no-cloning argument gives $Q=0$ for $p\ge1/12\approx0.0833$ [5]. The threshold therefore satisfies $0.064956\le p^{*}\le1/12$ , and evaluating $Q$ itself remains a major open problem [5].
General Pauli channels: hashing is achievable everywhere [1], and Agarwal et al. also raised the thresholds of two other Pauli families, the independent $X$ – $Z$ channel and the two-Pauli channel [5].

What would count as progress

A certified positive coherent information for the depolarizing channel at some $p>0.064956$ : an explicit $n$ -qubit input state with exact entries and a verifier using exact or interval arithmetic, as in [6].
A proof that $Q=0$ for the depolarizing channel at some $p<1/12$ , which would lower the upper end of the threshold window.
A sharper upper bound on $Q$ of the depolarizing channel at any $p$ , with a certificate that can be checked independently, such as a dual solution of a semidefinite program.

References

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, "Mixed-state entanglement and quantum error correction", Physical Review A 54, 3824–3851 (1996). doi:10.1103/PhysRevA.54.3824
D. P. DiVincenzo, P. W. Shor and J. A. Smolin, "Quantum-channel capacity of very noisy channels", Physical Review A 57, 830–839 (1998). doi:10.1103/PhysRevA.57.830
I. Devetak, "The private classical capacity and quantum capacity of a quantum channel", IEEE Transactions on Information Theory 51, 44–55 (2005). doi:10.1109/TIT.2004.839515
I. Devetak and P. W. Shor, "The capacity of a quantum channel for simultaneous transmission of classical and quantum information", Communications in Mathematical Physics 256, 287–303 (2005). doi:10.1007/s00220-005-1317-6
A. Agarwal, A. R. Kalra, S. Lee, D. Leung, L. Schaeffer, P. Sinha and G. Smith, "Enhanced quantum capacity thresholds from symmetry", preprint (2026). arXiv:2605.09138
A. Krohn-Grimberghe, "A certified lower bound on the quantum-capacity threshold of the depolarizing channel", preprint (2026). arXiv:2608.15870

Source

Imported from QIQCOP Zoo op_43d4aca67bd52554, curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.