Entanglement swapping for $X$ states demands threshold values

The basic entanglement-swapping scheme can be seen as a process which allows one to redistribute the Bell states properties between different pairs of a four-qubit system. We analyze a similar scheme by performing a Bell–von Neumann measurement over two local qubits, each one initially correlated through an $X$ state with a spatially distant qubit. This process swaps the $X$ feature without conditions, whereas the input entanglement is partially distributed in the four possible outcome states under certain conditions. Specifically, we obtain two threshold values for the entanglement of formation of the input $X$ states in order for the outcome states to be nonseparable.

Figure 1

EOF as functions of $\left|a\right|$ of the initial states (solid line) and of the outcome states $|{\ddot{\varphi }}_{A,B}^{\pm }\rangle$ (dashed line) and $|{\psi }_{A,B}^{\pm }\rangle$ (dotted line). The dot-dashed line corresponds to the average EOF $\overline{E}$ of the four possible outcome states.