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A definition of the Schmidt number of a state of an infinite dimensionalbipartite quantum system is given and properties of the corresponding family ofSchmidt classes are considered. The existence of states with a given Schmidtnumber such that any their countable convex decomposition does not contain purestates with finite Schmidt rank is established.  Partially entanglement breaking channels in infinite dimensions are studied.Several properties of these channels well known in finite dimensions aregeneralized to the infinite dimensional case. At the same time, the existenceof partially entanglement breaking channels (in particular, entanglementbreaking channels) such that all operators in any their Kraus representationshave infinite rank is proved.

The Schmidt number and partially entanglement breaking channels in infinite dimensions