{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"d336ec6f3259eb5e0ffb76e9bec8eaeb630af7be103a01d8f9c808d0041912ea","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":["ai:gpt-5.5"],"kind":"seed"},"parents":["a3da47a561710f538dca3b5b3fe360d3861dfd05220b716a479cc48f388159c6","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"7cfcee2128e674092e6dc0e3fda561874e5208d630f44dd474962eb621bacb0b","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":["ai:gpt-5.5"],"external_id":"qi.quantum-shannon-theory.depolarizing-capacity-threshold","kind":"well-posed","literature_status":"open","n":"2","parents":[],"plain":"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.","posed_since":"1996","precise":"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 $0 < p < 1/4$.","problem_ref":"a3da47a561710f538dca3b5b3fe360d3861dfd05220b716a479cc48f388159c6","references":"","settled_by":"A matching upper bound below $1/4$ and code construction, or an exact capacity formula.","status_note":"2026 preprints refine lower bounds on the threshold (e.g. arXiv:2608.15870, arXiv:2609.39747); $p*$ is unknown.","title":"Noise threshold for positive quantum capacity of the depolarizing 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