{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"f97887b7b76832c6eb35d6eb33b7eec6f2d6d083f72623245e0132026df534fe","created":"2026-10-03T07:18:10Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"c2ec860fd94b3723ecbf38c642ab344ce37a25d7eae23686a4011ee6a10e54f4","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"qi.sic-mub.mub-composite-max","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"In dimensions that are prime powers, the largest possible number of mutually unbiased bases is always reached. For every other dimension the maximum is unknown.","posed_since":"","precise":"Let $N(d)$ be the maximal number of MUBs in $C^d$, with $N(d) = d + 1$ for prime powers. Determine whether $N(d) < d + 1$ for every d that is not a prime power, and find $N(d)$ for $d = 10\\text{ and }12$. Lower bounds exceed the tensor-product bound $1 + \\operatorname{min}_i p_i^{a_i}$ in some dimensions, e.g. five MUBs in $d = 12$ (Cardenes Wuttig and Tindall, arXiv:2609.40311, 2026).","problem_ref":null,"references":"","settled_by":"Proofs of upper bounds below d + 1 together with matching constructions in specific composite dimensions.","status_note":"A September 2026 preprint gives constructions beyond the tensor-product bound in many composite dimensions.","title":"Maximum number of mutually unbiased bases in composite dimensions","topic_ref":"1bdfc0d09903d73034d6e6510c6d21f380d002328cb4a356dee914b540fb53cc"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"f40998596bac10f98640f4ea60b24897b7408f7ec3baa5aaeaece28c93fa5538","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"67d5db42dde234d0c68e7f43e8feae8803af3a91a2e70c2a5e6d79eec857ea89","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"S76l_nmaDwvHjQIDLQPXFc1eK9HUMUllSb72vi37wnrVuoEuojBu3jYh5L46qdW02STZQ8hD-Pp2DUAdRysVDQ"},"schema":"pubphys.envelope/1"},"record_hash":"f40998596bac10f98640f4ea60b24897b7408f7ec3baa5aaeaece28c93fa5538","leaf_index":2053}