Entanglement Swapping of Two Arbitrarily Degraded Entangled States

TL;DR

This paper provides a comprehensive analytical framework for entanglement swapping involving arbitrary partially mixed two-qubit states, crucial for quantum network operations and understanding entanglement degradation.

Contribution

It introduces a general analytical solution for entanglement swapping with arbitrary two-qubit states, including bounds on entanglement and relationships between initial and final state ranks.

Findings

Conservation of concurrence when swapping with a Bell state

Bounds on the concurrence of the resulting state

Relationship between initial and final state ranks

Abstract

We consider entanglement swapping, a key component of quantum network operations and entanglement distribution. Pure entangled states, which are the desired input to the swapping protocol, are typically mixed by environmental interactions causing a reduction in their degree of entanglement. Thus an understanding of entanglement swapping with partially mixed states is of importance. Here we present a general analytical solution for entanglement swapping of arbitrary two-qubit states. Our result provides a comprehensive method for analyzing entanglement swapping in quantum networks. First, we show that the concurrence of a partially mixed state is conserved when this state is swapped with a Bell state. Then, we find upper and lower bounds on the concurrence of the state resulting from entanglement swapping for various classes of input states. Finally, we determine a general relationship between the ranks of the initial states and the rank of the final state after swapping.