{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"9f0f0853de1bb3de9eea94d09b36d2cf1832a79f5fb6757bee230e05a8607ec4","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b","ef1018432235b6b1ee074807c1c8cbb18d6b543488fca1a92bd40e3d5752cd07"],"salt":"50e9adf25f4abaf5078bbe78e622ddf6e935a727d9d9b9548673d70f553c38c0","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":"2","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":"$Q(N) = \\lim_{n \\to \\infty} (1/n) \\max_\\rho I_c(\\rho, N^{(\\otimes\\ n)})$, 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":"ef1018432235b6b1ee074807c1c8cbb18d6b543488fca1a92bd40e3d5752cd07","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":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"e59bb2591f6cfb66d9dd3233877043f8b47faaf91f57c8c39d1e9ab5224e6d32","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"a8a027849b75ef6c5d172f7406028d4b3abb99437be93a4782da4ae3ed4ef0e9","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"JhVsqd2svWxUBxEDxaergs2EV6ek9kesuWNl_-p0yFZyNPGtOPt6HfiQodg28OXTJxXHricCpDbVJtkTfDmaCg"},"schema":"pubphys.envelope/1"},"record_hash":"e59bb2591f6cfb66d9dd3233877043f8b47faaf91f57c8c39d1e9ab5224e6d32","leaf_index":3746}