{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T03:40:09Z","timestamp":1756784409787,"version":"3.44.0"},"reference-count":21,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:p> We consider the compact convex set of all bipartite states of Schmidt number less than or equal to [Formula: see text], together with that of [Formula: see text]-blockpositive matrices of trace one, which play the roles of Schmidt number witnesses. In this note, we look for hyperplanes which support those convex sets and are perpendicular to a one parameter family through the maximally mixed state. We show that this is equivalent to determining the intervals for the dual objects on the one parameter family. We illustrate our results for the one parameter families including Werner states and isotropic states. Through the discussion, we give a simple decomposition of the separable Werner state into the sum of product states. <\/jats:p>","DOI":"10.1142\/s1230161225500088","type":"journal-article","created":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T03:06:23Z","timestamp":1756782383000},"source":"Crossref","is-referenced-by-count":0,"title":["Supporting Hyperplanes for Schmidt Numbers and Schmidt Number Witnesses"],"prefix":"10.1142","volume":"32","author":[{"given":"Kyung Hoon","family":"Han","sequence":"first","affiliation":[{"name":"Department of Data Science, The University of Suwon, Gyeonggi-do 445-743, Korea"}]},{"given":"Seung-Hyeok","family":"Kye","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Institute of Mathematics, Seoul National University, Seoul 151-742, Korea"}]}],"member":"219","published-online":{"date-parts":[[2025,9,3]]},"reference":[{"key":"S1230161225500088BIB001","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.73.032315"},{"key":"S1230161225500088BIB002","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/49\/8\/085301"},{"key":"S1230161225500088BIB003","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.66.062311"},{"key":"S1230161225500088BIB004","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.86.034301"},{"key":"S1230161225500088BIB006","doi-asserted-by":"publisher","DOI":"10.1016\/0034-4877(72)90011-0"},{"key":"S1230161225500088BIB007","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1996-010-x"},{"key":"S1230161225500088BIB008","first-page":"669","volume":"33","author":"Kye S.-H.","year":"1996","journal-title":"J. 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