Entanglement swapping for partially entangled qudits and the role of quantum complementarity

TL;DR

This paper generalizes entanglement swapping to partially entangled qudits, analyzes the process using quantum complementarity, and establishes bounds on the distributed entanglement, highlighting the role of local coherence and predictability.

Contribution

It extends entanglement swapping protocols to higher-dimensional qudits and links the process to quantum complementarity, providing bounds and insights into the role of local coherence.

Findings

Average distributed entanglement is bounded by initial entanglement.

Using states with zero local coherence simplifies the analysis.

Upper bounds on entanglement can be improved for qubits and qutrits.

Abstract

We extend the entanglement swapping protocol (ESP) to partially entangled qudit states and analyze the process within the framework of complete complementarity relations (CCRs). Building on previous results for qubits, we show that the average distributed entanglement between two parties via ESP is bounded above by the initial entanglement of one of the input pairs, and also by the product of the initial entanglements. Notably, we find that using initial states with vanishing local quantum coherence is sufficient to capture the essential features of the protocol, simplifying the analysis. By exploring the cases of qubits and qutrits, we observe that the upper bound on the average distributed entanglement -- expressed in terms of the product of the initial entanglements -- can be improved, and we conjecture what this tighter bound might be. Finally, we discuss the role of quantum complementarity in the ESP and show how local predictability constrains the entanglement that can be operationally distributed via ESP.