Concurrence and entanglement entropy of stochastic one-qubit maps

Explicit expressions for the concurrence of all positive and trace-preserving (stochastic) one-qubit maps are presented. Our method allows to construct the relevant convex roof patterns. We conclude that two-component optimal decompositions always exist. Our results can be transferred to 2xn-quantum systems 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 one-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.