  We analyze how a maximally entangled state of two-qubits (e.g., the singlet
$\psi_s$) is affected by action of local channels described by completely
positive maps $\cE$ . We analyze the concurrence and the purity of states
$\varrho_\cE=\cE\otimes\cI[\psi_s]$.Using the concurrence-{\it vs}-purity phase
diagram we characterize local channels $\cE$ by their action on the singlet
state $\psi_s$. We specify a region of the concurrence-{\it vs.}-purity diagram
that is achievable from the singlet state via the action of unital channels. We
show that even most general (including non-unital) local channels acting just
on a single qubit of the original singlet state cannot generate the maximally
entangled mixed states (MEMS). We study in detail various time evolutions of
the original singlet state induced by local Markovian semigroups. We show that
the decoherence process is represented in the concurrence-{\it vs.}-purity
diagram by a line that forms the lower bound of the achievable region for
unital maps. On the other hand, the depolarization process is represented by a
line that forms the upper bound of the region of maps induced by unital maps.
