  We give a mathematical formulation for the Choi-Jamiolkowski (CJ)
correspondence in the infinite-dimensional case in the form close to one used
in quantum information theory. We show that there is no need to use a limiting
procedure to define "unnormalized maximally entangled state" and the
corresponding analog of the Choi matrix since they can be defined rigorously
as, in general, nonclosable forms on an appropriate dense subspace. The
properties of these forms are discussed in Sec. 2. An important question is:
when the CJ form is given by a bounded operator. This is the case for
entanglement-breaking channels: we prove this in Sec. 3 along with a version of
a result of Wolf et al. characterizing CJ operators which correspond to such
channels by giving precise definitions of a separable operator and a relevant
integral. In Sec. 4 we obtain explicit expressions for CJ forms and operators
defining a general Bosonic Gaussian channel. In Sec. 5 we give a decomposition
of the Gaussian CJ form into product of the four principal types and a
necessary and sufficient condition for existence of the bounded CJ operator.
