2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/209469Explicit expressions for the concurrence of all positive and trace-preserving ("stochastic") 1-qubit maps are presented. By a new method we find the relevant convex roof pattern. We conclude that two component optimal decompositions always exist. Our results can be transferred to $2 \times n$-quantum systems providing the concurrence for all rank two density operators as well as a lower bound for their entanglement of formation.longer intro, small changes in textQuantum Physicstext