  Motivated by quantum thermodynamics we first investigate the notion of strict
positivity, that is, linear maps which map positive definite states to
something positive definite again. We show that strict positivity is decided by
the action on any full-rank state, and that the image of non-strictly positive
maps lives inside a lower-dimensional subalgebra. This implies that the
distance of such maps to the identity channel is lower bounded by one.
  The notion of strict positivity comes in handy when generalizing the
majorization ordering on real vectors with respect to a positive vector $d$ to
majorization on square matrices with respect to a positive definite matrix $D$.
For the two-dimensional case we give a characterization of this ordering via
finitely many trace norm inequalities and, moreover, investigate some of its
order properties. In particular it admits a unique minimal and a maximal
element. The latter is unique as well if and only if minimal eigenvalue of $D$
has multiplicity one.
