{"paper":{"title":"Defect of a unitary matrix","license":"","headline":"","cross_cats":["math.OA"],"primary_cat":"math.RA","authors_text":"Karol Zyczkowski, Wojciech Tadej","submitted_at":"2007-02-17T10:33:53Z","abstract_excerpt":"We analyze properties of a map B = f(U) sending a unitary matrix U of size N into a doubly stochastic matrix defined by B_{i,j} = |U_{i,j}|^2. For any U we define its DEFECT, determined by the dimensionality of the space being the image Df(T_U Unitaries) of the space T_U Unitaries tangent to the manifold of unitary matrices Unitaries at U, under the tangent map Df corresponding to f. The defect, equal to zero for a generic unitary matrix, gives an upper bound for the dimensionality of a smooth orbit (a manifold) of inequivalent unitary matrices V mapped into the same image, f(V) = f(U) = B, st"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702510","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}