Noise threshold for positive quantum capacity of the depolarizing channel
In plain words
The simplest noisy quantum channel replaces a qubit by a random error with probability p. The noise level above which it can no longer carry any quantum information at all is not known.
Precise statement
For the qubit depolarizing channel $D_p(\rho) = (1 - p) \rho + (p/3)(X \rho X + Y \rho Y + Z \rho Z)$, find p* with quantum capacity $Q(D_p) > 0$ if and only if p < p*. Known: $Q = 0$ for $p \ge 1/4$; the hashing bound is positive up to $p \sim 0.1893$, and degenerate codes push positivity above 0.19 (DiVincenzo, Shor and Smolin 1998; Smith and Smolin 2007), with further small improvements since. More generally, compute $Q(D_p)$ for any $0 < p < 1/4$.
What would settle it
A matching upper bound below $1/4$ and code construction, or an exact capacity formula.
Status in the literature
Unverified note
2026 preprints refine lower bounds on the threshold (e.g. arXiv:2608.15870, arXiv:2609.39747); $p*$ is unknown.