  To establish an entangled state of optimal fidelity between two distant
observers when the available quantum channel is noisy, is a central problem in
quantum information theory. We consider an instance of this problem for
two-qubit systems when only a single use of the channel and local
post-processing by trace preserving operations are allowed. We show that the
optimal fidelity is obtained only when part of an appropriate non-maximally
entangled state is transmitted through the channel. The entanglement of this
state can be vanishingly small when the channel becomes very noisy. Moreover,
in the optimal case no further local processing is required to enhance the
fidelity. We further show that local post-processing can enhance fidelity if
and only if the amount of noise is larger than a critical value and
entanglement of the transmitted state is bounded from below. A notable
consequence of these results is that the ordering of states under an
entanglement monotone can be reversed even when the states undergo the same
local interaction via a trace-preserving completely positive map.
