On the relation between states and maps in infinite dimensions

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 L-2(H-2, H-1) and the corresponding tensor products H-1 circle times 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 C : L-1(L-2(H-2, H-1)) -> L infinity(L(H-2), L-1(H-1)) from trace-class operators on L-2(H-2, H-1) (with the nuclear norm) into compact operators mapping the space of all bounded operators on H-2 into trace class operators on 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.