Infinite dimensional generalizations of Choi's theorem

In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi's characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A map $\mu$ is called a quantum channel, if it is trace preserving, and $\mu$ is called a quantum subchannel if it decreases the trace of a positive operator. We give simple neccesary and sufficient condtions for $\mu$ to be a quantum subchannel. We show that $\mu$ is a quantum subchannel if and only if it has Hellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.