  Explicit expressions for the concurrence of all positive and trace-preserving
("stochastic") 1-qubit maps are presented. We construct the relevant convex
roof patterns by a new method. We conclude that two component optimal
decompositions always exist. Our results can be transferred to
2xN-quantumsystems providing the concurrence for all rank two density operators
as well as lower and upper bounds for their entanglement of formation.
  We apply these results to a study of the entanglement entropy of 1-qubit
stochastic maps which preserve axial symmetry. Using analytic and numeric
results we analyze the bifurcation patterns appearing in the convex roof of
optimal decompositions and give results for the one-shot
(Holevo-Schumacher-Westmoreland) capacity of those maps.
