On the Relation between States and Maps in Infinite Dimensions

Abstract

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 * email: jagrab@impan.gov.pl † email: marek.kus@cft.edu.pl ‡ email: marmo@na.infn.it 1 generalized Choi Theorem. The natural framework is based on spaces of HilbertSchmidt operators L 2 (H 2 , H 1 ) and the corresponding tensor products H 1 ⊗ 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 traceclass operators. On the other hand, it is proven that there is a natural continu-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 operatornorm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

Stratified manifold of quantum states, actions of the complex special linear group

“…With a little effort [16,21,29,30,31] it is possible to prove that the orbits of SL(H) on S (H) by means of the action given in equation (32) are labelled by the rank of the quantum states they contain, and that they are differential manifolds. We will denote these orbits as S k (H), where k denotes the rank of the quantum states in S k (H).…”

“…Our understanding of the geometry of the space of quantum states is in constant evolution and there are different fields of application in which it is possible to use the knowledge we gain. For instance, geometrical ideas have been successfully exploited when addressing the foundations of quantum mechanics [4,7,8,10,13,20,22,23,25,31,35,40], quantum information theory [5,15,19,27,36,39,43,45,54], quantum dynamics [9,12,14,16,17,18,24], entanglement theory [3,6,11,29,30,34,48,49].…”

Geometry of quantum dynamics in infinite-dimensional Hilbert space

“…Since Hilbert-Schmidt operators can be interpreted as elements of the Hilbert tensor product H ⊗ H * , where the dual H * can be identified with H with respect to the anti-linear isomorphism (cf. [26])…”

Differential calculus on manifolds with boundary applications

“…In what follows we are going to consider the geometry of D A according to [6,26,27,2]. Then, considering the geometrization of the dynamics of open quantum systems, we will argue that a notion of vector fields (derivations) transversal to the strata is needed in order to describe physically interesting processes such as decoherence, for which the rank of the states is not preserved.…”

“…We shall consider only the case in which A is a finite-dimensional simple C * -algebra with unity, in which case it is necessarily isomorphic to the set B(H) of bounded linear operators on a complex Hilbert space H of dimension (dim(H)) 2 = dim(A) = N (see [28]). In this case, O A becomes isomorphic to the set of self-adjoint linear operators on H. We shall limit ourselves to a quick revision of the geometry of O * A and D A , we refer to [6,26,27,2] for a detailed discussion.…”