Quantum capacity of the qubit depolarizing channel

The depolarizing channel is a canonical simple noise model whose ordinary quantum capacity is still unknown. The parameter $q$ here is the total nonidentity Pauli error probability; the probability of each individual Pauli error is $p=q/3$. Other definitions of “depolarizing probability” differ, so numerical thresholds must state their convention.

Antidegradability gives zero quantum capacity for $q\in[1/4,3/4]$. Achievable rates from coherent information and degenerate quantum codes give lower bounds, but optimization over a single use is not generally the capacity. The residual gap already poses a sharp problem for a two-dimensional channel.

Preprint progress (August 2026): Krohn-Grimberghe supplies an exact-arithmetic certificate claiming positive coherent information for a 45-copy rank-two input at per-Pauli noise $p=0.064956$, or total error $q=0.194868$. If verified, this certifies $q_*\geq0.194868$; it is not an exact capacity formula or the threshold itself. The certificate is designed to permit independent checking, which this catalogue does not claim to have performed.

This benchmark was promoted from C1 to keep the precise threshold question and its noise normalization visible. Unassisted vanishing-error capacity is intended; two-way, zero-error and restricted-code capacities are different quantities.

Source disclosures

Krohn-Grimberghe reports an AI-driven numerical search and AI assistance with code, calculations, mechanical proof derivations and drafting under author direction. The author takes responsibility; the verification chain is intended to be independent of witness discovery. Source

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: Exact-arithmetic positive coherent-information certificate for a 45-copy rank-two input at total Pauli error q=0.194868; not an exact threshold. Evidence