{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"d741d7495757702cd42bdd695a1b95e0b571a8b06c4b2b9c40d616620b6a65b6","created":"2026-10-03T07:18:10Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"5fb5a05129031816eaaa3cae847ec6c27d755c53d74102066693cef5749afb2b","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"qi.quantum-shannon-theory.quantum-capacity-computability","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"The quantum capacity of a channel is defined by a limit over ever more channel uses, and some channels show positive capacity only after arbitrarily many uses. Whether any algorithm can compute the capacity to a given accuracy, or even decide whether it is zero, is unknown.","posed_since":"2015","precise":"$Q(N) = \\lim_{n \\to \\infty} (1/n) \\max_\\rho I_c(\\rho, N^{(\\mathrm{tensor}\\ n)})$, with $I_c$ the coherent information. Is there an algorithm that, given a channel N with rational Kraus operators and $\\epsilon > 0$, outputs $Q(N)$ to within $\\epsilon$, or that decides whether $Q(N) > 0$? Cubitt et al. (Nat. Commun. 6, 7739, 2015, arXiv:1408.5115) showed that for every n there are channels with zero n-use coherent information and positive capacity.","problem_ref":null,"references":"","settled_by":"A convergent algorithm with an explicit error bound for $Q(N)$, or a reduction from an undecidable problem to deciding $Q(N) > 0$.","status_note":"","title":"Is the quantum capacity of a channel computable","topic_ref":"66c5088d9a8eed6c4b6c5d401128245d915bf032a0cd5d9569f7f404a77d69c8"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"ef1018432235b6b1ee074807c1c8cbb18d6b543488fca1a92bd40e3d5752cd07","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"ff669236e69af8523f07e5e3d60e4c4dea727f95cd1885fec87456b1822faf75","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"VOaaXyTFjD1QD7MEMFltqemB2sdlPLMPs3fFQOmLkXNHWjTFn8TjylrOoeXAyQH2qETLTX3E65uG3sNOaQYwDQ"},"schema":"pubphys.envelope/1"},"record_hash":"ef1018432235b6b1ee074807c1c8cbb18d6b543488fca1a92bd40e3d5752cd07","leaf_index":2037}