{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T17:24:56Z","timestamp":1778693096265,"version":"3.51.4"},"reference-count":24,"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> Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper. <\/jats:p>","DOI":"10.1007\/s11080-007-9061-3","type":"journal-article","created":{"date-parts":[[2007,12,10]],"date-time":"2007-12-10T15:24:59Z","timestamp":1197300299000},"page":"355-370","source":"Crossref","is-referenced-by-count":14,"title":["On the Relation between States and Maps in Infinite Dimensions"],"prefix":"10.1142","volume":"14","author":[{"given":"Janusz","family":"Grabowski","sequence":"first","affiliation":[{"name":"Polish Academy of Sciences, Institute of Mathematics, \u015aniadeckich 8, P.O. Box 21, 00\u2013956 Warszawa, Poland"}]},{"given":"Marek","family":"Ku\u015b","sequence":"additional","affiliation":[{"name":"Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik\u00f3w 32\/46, 02\u2013668 Warszawa, Poland"}]},{"given":"Giuseppe","family":"Marmo","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Fisiche, Universit\u00e0 \u201cFederico II\u201d di Napoli and Istituto Nazionale di Flsica Nucleare, Sezione di Napoli, Complesso Universitario di Monte Sant Angelo, Via Cintia, 1-80126 Napoli, Italy"}]}],"member":"219","published-online":{"date-parts":[[2012,4,17]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01491891"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.47.777"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysicsPhysiqueFizika.1.195"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.38.447"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04209-0"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/38\/47\/011"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/s11080-006-9013-3"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.2307\/1969387"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-97306-2"},{"key":"rf11","first-page":"885","volume":"6","author":"Gleason A.","journal-title":"J. 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