Quantification of Entanglement of Teleportation in Arbitrary Dimensions

TL;DR

This paper investigates the relationship between entanglement measures and teleportation fidelity in arbitrary-dimensional bipartite states, establishing bounds and measures to identify states useful for teleportation, especially considering Schmidt rank and PPT states.

Contribution

It introduces new bounds and measures linking entanglement and teleportation fidelity, considering Schmidt rank and PPT states, to better identify states suitable for quantum teleportation.

Findings

Derived bounds for entanglement measures in terms of teleportation fidelity.

Established relations between Schmidt rank and teleportation usefulness.

Applied bounds to two-qutrit and two-qubit systems.

Abstract

We study bipartite entangled states in arbitrary dimensions and obtain different bounds for the entanglement measures in terms of teleportation fidelity. We find that there is a simple relation between negativity and teleportation fidelity for pure states but for mixed states, an upper bound is obtained for negativity in terms of teleportation fidelity using convex-roof extension negativity (CREN). However, with this it is not clear how to distinguish betweeen states useful for teleportation and positive partial transpose (PPT) entangled states. Further, there exists a strong conjecture in the literature that all PPT entangled states, in 3 \times 3 systems, have Schmidt rank two. This motivates us to develop measures capable of identifying states useful for teleportation and dependent on the Schmidt number. We thus establish various relations between teleportation fidelity and entanglement measures depending upon Schmidt rank of the states. These relations and bounds help us to determine the amount of entanglement required for teleportation, which we call the ``Entanglement of Teleportation''. These bounds are used to determine the teleportation fidelity as well as the entanglement required for teleportation of states modeled by a two qutrit mixed system, as well as two qubit open quantum systems.