On the Relation between States and Maps in Infinite Dimensions

Relations between states and maps, which are known for quantum systems in finitedimensional 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.