  We apply random matrix and free probability techniques to the study of linear
maps of interest in quantum information theory. Random quantum channels have
already been widely investigated with spectacular success. Here, we are
interested in more general maps, asking only for $k$-positivity instead of the
complete positivity required of quantum channels. Unlike the theory of
completely positive maps, the theory of $k$-positive maps is far from being
completely understood, and our techniques give many new parametrized families
of such maps. We also establish a conceptual link with free probability theory,
and show that our constructions can be obtained to some extent without random
techniques in the setup of free products of von Neumann algebras. Finally, we
study the properties of our examples and show that for some parameters, they
are indecomposable. In particular, they can be used to detect the presence of
entanglement missed by the partial transposition test, that is, PPT
entanglement. As an application, we considerably refine our understanding of
PPT states in the case where one of the spaces is large whereas the other one
remains small.
