Entanglement sharing via qudit channels: Nonmaximally entangled states may be necessary for one-shot optimal singlet fraction and negativity

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.