## 4 Multi-Hop Bidirectional Quantum Communication
### Example of Multi-Hop Bidirectional Quantum Communication  
An example is presented to better illustrate the whole process of multi-hop bidirectional quantum communication. Assume that two nodes named \(N_{1}\) and \(N_{5}\) intend to transmit two qubits \(|\chi\rangle_{a}\) and \(|\chi\rangle_{b}\) to each other. We might as well suppose that three intermediate nodes \(N_{2}\), \(N_{3}\) and \(N_{4}\) are involved in the quantum path '\(N_{1}\to N_{2}\to N_{3}\to N_{4}\to N_{5}\)', as shown in Figure 6.  
Figure 5: The composite GHZ-GHZ entanglement channel in the \(n\)-hop quantum communication case.  
Figure 6: An example of multi-hop bidirectional quantum communication.  
Assume that the composite GHZ-GHZ states \(\left|\psi^{1}\right\rangle_{N_{1}^{1}N_{2}^{1}N_{3}^{2}}-\left|\psi^{1}\right\rangle _{N_{2}^{2}N_{1}^{3}N_{2}^{3}}\), \(\left|\psi^{5}\right\rangle_{N_{2}^{4}N_{3}^{1}N_{2}^{5}}-\left|\psi^{5}\right\rangle _{N_{3}^{2}N_{2}^{6}N_{3}^{3}}\), \(\left|\psi^{4}\right\rangle_{N_{3}^{4}N_{4}^{1}N_{3}^{5}}-\left|\psi^{4}\right\rangle _{N_{4}^{2}N_{3}^{6}N_{4}^{3}}\), and \(\left|\psi^{8}\right\rangle_{N_{4}^{4}N_{5}^{1}N_{4}^{5}}-\left|\psi^{8}\right \rangle_{N_{2}^{2}N_{4}^{6}N_{5}^{3}}\) were previously shared on the quantum path. To implement the qubit transmission from \(N_{1}\) to \(N_{5}\), \(N_{1}\) and all the intermediate nodes perform quantum measurements simultaneously, then they send their measurement results and the type of the composite GHZ-GHZ states previously shared between themselves and the next-hop nodes to the node \(N_{5}\) through classical channels. Assume the node \(N_{5}\) receives a group of measurement results \(\left|011\right\rangle_{sN_{1}^{1}N_{2}^{1}}\), \(\left|100\right\rangle_{N_{2}^{1}N_{3}^{1}N_{4}^{5}}\), \(\left|001\right\rangle_{N_{3}^{1}N_{3}^{4}N_{3}^{5}}\), and \(\left|001\right\rangle_{N_{4}^{1}N_{4}^{4}N_{4}^{5}}\) from \(N_{1}\), \(N_{2}\), \(N_{3}\), \(N_{4}\) respectively; the unitary operation required for recovering the original qubit locally is given by  
\[U=\prod_{i=1}^{4}T_{i}=X\cdot I\cdot X\cdot ZX=ZX \tag{11}\]  
where \(T_{1}=X\), \(T_{2}=I\), \(T_{3}=X\), and \(T_{4}=ZX\) are determined according to Table 2.  
When it comes to the qubit transmission from \(N_{5}\) to \(N_{1}\), \(N_{5}\) and all the intermediate nodes perform quantum measurements simultaneously, then they send their measurement results and the type of the composite GHZ-GHZ state previously shared between themselves and the next-hop node to the node \(N_{1}\) through classical channels. Assume the node \(N_{1}\) receives a group of measurement results \(\left|011\right\rangle_{N_{2}^{6}N_{2}^{2}N_{2}^{3}}\), \(\left|100\right\rangle_{N_{5}^{6}N_{2}^{2}N_{3}^{3}}\), \(\left|101\right\rangle_{N_{4}^{6}N_{4}^{2}N_{4}^{3}}\), and \(\left|110\right\rangle_{dN_{5}^{2}N_{5}^{3}}\) from \(N_{2}\), \(N_{3}\), \(N_{4}\), and \(N_{5}\) respectively; the unitary operation required for recovering the original qubit locally is given by  
\[V=\prod_{j=4}^{1}C_{j+1}=I\cdot ZX\cdot I\cdot X=Z \tag{12}\]  
where \(C_{5}=I\), \(C_{4}=ZX\), \(C_{3}=I\), and \(C_{2}=X\) are determined according to Table 2.