On the relation between states and maps in infinite dimensions

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dc.creatorGrabowski, Janusz
dc.creatorKus, Marek
dc.creatorMarmo, Giuseppe
dc.date2007-06-18
dc.date.accessioned2026-07-07T08:10:50Z
dc.date.available2026-07-07T08:10:50Z
dc.descriptionRelations 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 $\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1)$ and the corresponding tensor products $\mathcal{H}_1\otimes\mathcal{H}_2^*$ 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 $\mathcal{C}:\mathcal{L}_1(\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1))\to \mathcal{L}_\infty(\mathcal{L}(\mathcal{H}_2),\mathcal{L}_1(\mathcal{H}_1))$ from trace-class operators on $\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1)$ (with the nuclear norm) into compact operators mapping the space of all bounded operators on $\mathcal{H}_2$ into trace class operators on $\mathcal{H}_1$ (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.
dc.description19 pages
dc.identifierhttps://arxiv.org/abs/0706.2617
dc.identifierhttp://arxiv.org/abs/0706.2617
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/131936
dc.subjectMathematical Physics
dc.titleOn the relation between states and maps in infinite dimensions
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