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

TL;DR

This paper investigates entanglement sharing over noisy quantum channels, revealing that nonmaximally entangled states are often necessary for optimal entanglement measures in higher dimensions, challenging previous assumptions.

Contribution

It demonstrates that in higher dimensions, optimal entanglement cannot always be achieved with maximally entangled states, and the known qubit formulas do not generalize.

Findings

Nonmaximally entangled states are required for optimal singlet fraction in certain channels.

The formula for one-shot optimal singlet fraction in qubits does not extend to higher dimensions.

Entanglement ordering may not be preserved under LOCC in all finite dimensions.

Abstract

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.