Quantum capacity of a qubit Pauli channel

Problem

What is the unassisted quantum capacity of a general qubit Pauli channel? Let \(\mathbf p=(p_I,p_X,p_Y,p_Z)\) be a probability vector, with \(p_i\geq0\) for \(i\in\{I,X,Y,Z\}\) and \(\sum_{i\in\{I,X,Y,Z\}}p_i=1\). Define

\begin{equation} \Lambda_{\mathbf p}(\rho) =p_I\rho+p_XX\rho X+p_YY\rho Y+p_ZZ\rho Z, \tag{1} \end{equation}

where \(\rho\) is a qubit density operator and \(X,Y,Z\) are the Pauli matrices. The unassisted quantum capacity \(\mathcal Q(\Lambda_{\mathbf p})\) is the supremum of asymptotic qubit transmission rates achievable with vanishing error over independent channel uses. Arbitrary block encodings and joint decoding are allowed, without classical communication assistance or preshared entanglement. Equivalently,

\begin{equation} \mathcal Q(\Lambda_{\mathbf p}) =\sup_{n\geq1}\frac1n\max_{\rho_{A^n}} I_c\!\left(\rho_{A^n},\Lambda_{\mathbf p}^{\otimes n}\right), \tag{2} \end{equation}

where \(\rho_{A^n}\) ranges over all states of \(n\) input qubits [Dev05], Proposition 7. The coherent information in Eq. (2) is

\begin{equation} I_c(\rho,\mathcal N) =S(\mathcal N(\rho))-S(\mathcal N^c(\rho)), \tag{3} \end{equation}

where \(\mathcal N^c\) is a complementary channel and \(S(\sigma)=-\operatorname{Tr}(\sigma\log_2\sigma)\) is the von Neumann entropy used in Eq. (3). Determine \(\mathcal Q(\Lambda_{\mathbf p})\) in Eq. (2) for the channel of Eq. (1) throughout the probability simplex, including the exact boundary between zero and positive capacity.

Source

The exact-capacity question is implicit in the hashing lower bound of Bennett et al. and the strict degenerate-code improvement of DiVincenzo, Shor, and Smolin, which together leave the general Pauli-channel capacity undetermined [BDSW96], [DSS98].

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The hashing construction gives the achievable lower bound

\begin{equation} \mathcal{Q}(\Lambda_{\mathbf p})\ge \max\{0,1-H(\mathbf p)\}, \qquad H(\mathbf p):=-\sum_{i\in\{I,X,Y,Z\}}p_i\log_2p_i, \qquad 0\log_2 0:=0. \tag{4} \end{equation}

The quantity \(1-H(\mathbf p)\) in Eq. (4) equals \(I_c(I/2,\Lambda_{\mathbf p})\), the coherent information at the maximally mixed input (often called the symmetric coherent information), rather than the optimized one-shot coherent information in general [BDSW96], [Dev05].
The coherent-information rate in Eq. (4) is achieved by efficient quantum polar codes for qubit Pauli channels. The first construction uses preshared entanglement (with zero consumption for sufficiently low noise), while a later construction removes the need for preshared entanglement [RDR12], [RSDR15].
In the depolarizing case, write \(p_I=f\) and \(p_X=p_Y=p_Z=(1-f)/3\), and let \(r:=(1-f)/3\) be the per-Pauli error probability. Krohn-Grimberghe constructed an explicit rank-two, permutation-symmetric input across \(45\) channel uses and certified in exact rational arithmetic that its coherent information is positive at \(r=16239/250000=0.064956\). Monotonicity under channel post-processing therefore gives \(\mathcal Q(\Lambda_{\mathbf p})>0\) whenever \(f\ge201283/250000=0.805132\). In particular, throughout \(0.805132\le f<0.81071\) the capacity is positive although the hashing expression in Eq. (4) is negative. Table 1 of the paper lists the preceding best printed positivity point as \(r=0.064657\), obtained by Agarwal et al. through symmetry-reduced coherent-information optimization [AKL+26]. The new point is machine-checkable but is not claimed to be the exact capacity threshold [KG26].
For the qubit depolarizing subfamily, numerical optimization over channel extensions parameterized by the Stiefel manifold produces upper bounds on \(\mathcal Q(\Lambda_{\mathbf p})\) that strictly improve the previously best flagged-extension bounds over the high-noise interval studied. The optimized bounds do not meet the known achievable rates, so they narrow the capacity gap without determining the capacity. The same work also proves that its amortized channel coherent information equals the ordinary one-shot channel coherent information for every channel, ruling out that amortization as a route to a stronger capacity lower bound [ZMFW25].
Kondra et al. obtained an efficiently computable, all-code exponential strong-converse bound for every full-support qubit Pauli channel. For the depolarizing convention above, with total error \(\delta:=1-f\), their bound reduces to \(\mathcal Q(\Lambda_{\mathbf p})\le\max\{1-4\delta,0\}\); it is stricter than the regularized channel Rains bound for \(\delta\gtrsim0.15642\) and vanishes at the antidegradability threshold \(\delta=1/4\). This sharpens the high-noise converse but does not determine the capacity below that threshold [KBK+26].

Comment

The exact quantum capacity remains unknown for a general qubit Pauli channel; the hashing rate is an achievable lower bound and need not be the capacity.

References

[BDSW96]: 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 arXiv
[Dev05]: I. Devetak, “The Private Classical Capacity and Quantum Capacity of a Quantum Channel,” IEEE Transactions on Information Theory 51, 44–55 (2005).DOI arXiv
[RDR12]: J. M. Renes, F. Dupuis, and R. Renner, “Efficient Polar Coding of Quantum Information,” Physical Review Letters 109, 050504 (2012).DOI arXiv
[RSDR15]: J. M. Renes, D. Sutter, F. Dupuis, and R. Renner, “Efficient Quantum Polar Codes Requiring No Preshared Entanglement,” IEEE Transactions on Information Theory 61, 6395–6414 (2015).DOI arXiv
[DSS98]: 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 arXiv
[AKL+26]: A. Agarwal, A. R. Kalra, S. Lee, D. Leung, L. Schaeffer, P. Sinha, and G. Smith, “Enhanced Quantum Capacity Thresholds from Symmetry,” arXiv preprint (2026).arXiv
[KG26]: A. Krohn-Grimberghe, “A Certified Lower Bound on the Quantum-Capacity Threshold of the Depolarizing Channel,” arXiv preprint (2026).arXiv
[ZMFW25]: C. Zhu, H. Mao, K. Fang, and X. Wang, “Geometric Optimization for Quantum Communication,” arXiv preprint (2025).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

Quantum capacity of a qubit Pauli channel

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