Entanglement swapping theory and beyond

TL;DR

This paper develops a comprehensive theoretical framework for entanglement swapping involving pure, mixed, and maximally entangled states, extending to multi-state chains and providing new proofs for previous results.

Contribution

It introduces generalized entanglement swapping theories for pure, mixed, and multi-level states, including chains and new proof techniques, broadening the understanding of quantum entanglement manipulation.

Findings

Generalized entanglement swapping for pure and mixed states.

Proposed entanglement swapping chains with mathematical induction proof.

Extended theory to multi-level and multi-state entanglement scenarios.

Abstract

We focus on the general theory of entanglement swapping, including entanglement swapping of pure and mixed states. We also study the theory of entangled swapping chains, which can be regarded as an application of entangled swapping. For maximally entangled states, we consider the entanglement swapping of 2-level maximally entangled states without limiting each subsystem to be in the same basis. We further consider the entanglement swapping between d-level maximally entangled states, which is realized by performing a joint measurement on the particles that contain the first particle of the selected entangled states and the particles without involving the first particle in other entangled states. For the entanglement swapping of general pure state, we generalize the case of two bipartite general pure states to multi-state cases. Besides, we propose the entanglement swapping between bipartite general pure states and maximally entangled states. For entanglement swapping chains, we propose the entanglement swapping chains for maximally entangled states, which is realized by performing joint measurements on multiple particles in each state, we use mathematical induction to prove the results of entanglement swapping chains of maximally entangled states and that of general pure states. Moreover, we study the entanglement swapping and entanglement swapping chains of mixed states, including X states and mixed maximally entangled states. Finally, we provide a new proof for our previous work [2022, Physica A, 585, 126400].