  In this paper we give a general integral representation for separable states
in the tensor product of infinite dimensional Hilbert spaces and provide the
first example of separable states that are not countably decomposable. We also
prove the structure theorem for the quantum communication channels that are
entanglement-breaking, generalizing the finite-dimensional result of M.
Horodecki, Ruskai and Shor. In the finite dimensional case such channels can be
characterized as having the
  Kraus representation with operators of rank 1. The above example implies
existence of infinite-dimensional entanglement-breaking channels having no such
representation.
