{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"62efdaa345856195c9554d3e9564b20edc995289b070c44abba2de25f6373c19","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b","f40998596bac10f98640f4ea60b24897b7408f7ec3baa5aaeaece28c93fa5538"],"salt":"f5ac6efbe62e2bedcbe9e2b177ca912b11bb7abb94bf1bc83c29e1aaa9c72a6a","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":"2","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 $1 + \\min_i p_i^{a_i}$ in some dimensions, e.g. five MUBs in $d = 12$ (Cardenes Wuttig and Tindall, arXiv:2609.40311, 2026).","problem_ref":"f40998596bac10f98640f4ea60b24897b7408f7ec3baa5aaeaece28c93fa5538","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 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