{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T14:32:59Z","timestamp":1779373979130,"version":"3.53.1"},"reference-count":20,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2006,3]]},"abstract":"<jats:p> Recently, a powerful separability criterion was introduced by O. Rudolf in [5] and by K. Chen et al. in [6] \u2014 basing on realignment of elements of density matrix. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely tri-parite entanglement. We generalize it by introducing a family of so called permutation separability criteria for multipartite states. Namely, any permutation of indices of density matrix written in product basis leads to a separability criterion. Partial transpose and realignment criterion are special cases of permutation criteria. <\/jats:p>","DOI":"10.1007\/s11080-006-7271-8","type":"journal-article","created":{"date-parts":[[2006,3,4]],"date-time":"2006-03-04T01:37:46Z","timestamp":1141436266000},"page":"103-111","source":"Crossref","is-referenced-by-count":92,"title":["Separability of Mixed Quantum States: Linear Contractions and Permutation Criteria"],"prefix":"10.1142","volume":"13","author":[{"given":"Micha\u0142","family":"Horodecki","sequence":"first","affiliation":[{"name":"Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, 80-952 Gda\u0144sk, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pawe\u0142","family":"Horodecki","sequence":"additional","affiliation":[{"name":"Faculty of Applied Physics and Mathematics, Technical University of Gda\u0144sk, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ryszard","family":"Horodecki","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics and Astrophysics, University of Gda\u0144sk, 80-952 Gda\u0144sk, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2012,4,17]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.40.4277"},{"key":"rf2","volume-title":"Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments","author":"Alber G.","year":"2001"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1080\/09500340110105975"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.77.1413"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.67.032312"},{"key":"rf6","first-page":"193","volume":"3","author":"Chen K.","journal-title":"Quantum Inf. 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