  We introduce the notion of one-sided mapping cones of positive linear maps
between matrix algebras. These are convex cones of maps that are invariant
under compositions by completely positive maps from either the left or right
side. The duals of such convex cones can be characterized in terms of
ampliation maps, which can also be used to characterize many notions from
quantum information theory---such as separability, entanglement-breaking maps,
Schmidt numbers, as well as decomposable maps and $k$-positive maps in
functional analysis. In fact, such characterizations hold if and only if the
involved cone is a one-sided mapping cone. Through this analysis, we obtain
mapping properties for compositions of cones from which we also obtain several
equivalent statements of the PPT (positive partial transpose) square
conjecture.
