  We consider the problem of establishing entangled states of optimal singlet
fraction and negativity between two remote parties for every use of a noisy
quantum channel and trace-preserving LOCC under the assumption that the parties
do not share prior correlations. We show that for a family of quantum channels
in every finite dimension $d\geq3$, one-shot optimal singlet fraction and
entanglement negativity are attained only with appropriate nonmaximally
entangled states. We further show that the generalization of the formula that
exactly computes one-shot optimal singlet fraction for qubit channels does not
hold in general in higher dimensions. A consequence of our results is that the
ordering of entangled states in all finite dimensions may not be preserved
under trace-preserving LOCC.
