{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T06:05:59Z","timestamp":1784700359680,"version":"3.55.0"},"reference-count":14,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","funder":[{"DOI":"10.13039\/501100001809","name":"NNSF of China","doi-asserted-by":"crossref","award":["12471427"],"award-info":[{"award-number":["12471427"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["ZG216S2110"],"award-info":[{"award-number":["ZG216S2110"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2026,6]]},"abstract":"<jats:p>The partial transpose map is a linear map widely used in quantum information theory. We study the equality condition for a matrix inequality generated by partial transpose, namely [Formula: see text], where [Formula: see text]\u2019s and [Formula: see text]\u2019s are respectively the matrices of the same size, and [Formula: see text] is the Schmidt rank. We find explicit conditions that are equivalent to equality when the [Formula: see text]\u2019s are column or row vectors, or [Formula: see text] matrices. For the case where the Schmidt rank equals the dimension of [Formula: see text], we extend the results from [Formula: see text] matrices to square matrices, and further to rectangular matrices. In detail, we show that [Formula: see text] is locally equivalent to an elegant block-diagonal form consisting solely of identity and zero matrices. We also study the general case for [Formula: see text], and it turns out that the key is to characterize the expression of matrices [Formula: see text]\u2019s and [Formula: see text]\u2019s.<\/jats:p>","DOI":"10.1142\/s1230161226500071","type":"journal-article","created":{"date-parts":[[2026,7,22]],"date-time":"2026-07-22T05:50:06Z","timestamp":1784699406000},"source":"Crossref","is-referenced-by-count":0,"title":["Equality Condition for a Matrix Inequality by Partial Transpose"],"prefix":"10.1142","volume":"33","author":[{"given":"Nalan","family":"Wang","sequence":"first","affiliation":[{"name":"LMIB (Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University, Beijing 100191, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lin","family":"Chen","sequence":"additional","affiliation":[{"name":"LMIB (Beihang University), Ministry of Education, and School of Mathematical Sciences, Beihang University, Beijing 100191, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2026,7,20]]},"reference":[{"key":"S1230161226500071BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2014.03.035"},{"key":"S1230161226500071BIB002","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2012.2185036"},{"key":"S1230161226500071BIB003","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.78.022318"},{"issue":"28","key":"S1230161226500071BIB004","first-page":"1213","volume":"44","author":"Chen L.","year":"2012","journal-title":"Journal of Physics A Mathematical and Theoretical"},{"key":"S1230161226500071BIB005","doi-asserted-by":"crossref","unstructured":"L. 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