{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T07:52:44Z","timestamp":1712562764521},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2007,12]]},"abstract":"<jats:p> Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and, more generally, as witnesses for states having Schmidt number not greater than k. Such maps and witnesses are related to k-positive maps, and their properties may be investigated by making use of the Jamio\u0142kowski isomorphism. In this article we review the properties of this isomorphism, noting that there are actually two related mappings bearing that name. As a new result, we give a simplified proof for the correspondence between vectors having Schmidt number k and k-positive maps and thus for the Jamio\u0142kowski criterion for complete positivity. Another consequence is a special case of a result by Choi, namely that k-positivity implies complete positivity, if k is the dimension of the smaller one of the Hilbert spaces on which the operators act. <\/jats:p>","DOI":"10.1007\/s11080-007-9062-2","type":"journal-article","created":{"date-parts":[[2007,12,10]],"date-time":"2007-12-10T15:24:59Z","timestamp":1197300299000},"page":"371-378","source":"Crossref","is-referenced-by-count":12,"title":["The Jamio\u0142kowski Isomorphism and a Simplified Proof for the Correspondence Between Vectors Having Schmidt Number k and k-Positive Maps"],"prefix":"10.1142","volume":"14","author":[{"given":"Kedar S.","family":"Ranade","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Angewandte Physik, Technische Universit\u00e4t Darmstadt Hochschulstra\u00dfe 4a, 64289 Darmstadt, Germany"}]},{"given":"Mazhar","family":"Ali","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte Physik, Technische Universit\u00e4t Darmstadt Hochschulstra\u00dfe 4a, 64289 Darmstadt, Germany"}]}],"member":"219","published-online":{"date-parts":[[2012,4,17]]},"reference":[{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.60.898"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1972-044-5"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(75)90075-0"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.71.032332"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.59.4206"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0034-4877(72)90011-0"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1002\/9783527618637.ch5"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01609396"},{"key":"rf11","volume-title":"Quantum Computation and Quantum Information","author":"Nielsen M. A.","year":"2000"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.77.1413"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1967.23.129"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1007\/s11080-005-0486-2"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.63.050301"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.61.040301"}],"container-title":["Open Systems &amp; Information Dynamics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1007\/s11080-007-9062-2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T19:59:06Z","timestamp":1565121546000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1007\/s11080-007-9062-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,12]]},"references-count":14,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2012,4,17]]},"published-print":{"date-parts":[[2007,12]]}},"alternative-id":["10.1007\/s11080-007-9062-2"],"URL":"https:\/\/doi.org\/10.1007\/s11080-007-9062-2","relation":{},"ISSN":["1230-1612","1793-7191"],"issn-type":[{"value":"1230-1612","type":"print"},{"value":"1793-7191","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,12]]}}}