  Maximally entangled states--a resource for quantum information
processing--can only be shared through noiseless quantum channels, whereas in
practice channels are noisy. Here we ask: Given a noisy quantum channel, what
is the maximum attainable purity (measured by singlet fraction) of shared
entanglement for single channel use and local trace preserving operations? We
find an exact formula of the maximum singlet fraction attainable for a qubit
channel and give an explicit protocol to achieve the optimal value. The
protocol distinguishes between unital and nonunital channels and requires no
local post-processing. In particular, the optimal singlet fraction is achieved
by transmitting part of an appropriate pure entangled state, which is maximally
entangled if and only if the channel is unital. A linear function of the
optimal singlet fraction is also shown to be an upper bound on the distillable
entanglement of the mixed state dual to the channel.
