  Positive maps which are not completely positive are used in quantum
information theory as witnesses for convex sets of states, in particular as
entanglement witnesses and more generally as witnesses for states having
Schmidt number not greater than k. It is known that such witnesses are related
to k-positive maps. In this article we propose a new proof for the
correspondence between vectors having Schmidt number k and k-positive maps
using Jamiolkowski's criterion for positivity of linear maps; to this aim, we
also investigate the precise notion of the term "Jamiolkowski isomorphism". As
consequences of our proof we get the Jamiolkowski criterion for complete
positivity, and we find a special case of a result by Choi, namely that
k-positivity implies complete positivity, if k is the dimension of the smaller
one of the Hilbert spaces on which the operators act.
