## 2 One-Hop Bidirectional Quantum Communication  
In this section, we investigate one-hop bidirectional quantum communication through the composite GHZ-GHZ states. Eight types of GHZ states used in quantum communication are given by  
\[ \begin{array}{l}\left|\psi^{1}\right\rangle=\frac{1}{\sqrt{2}}(\left|000 \right\rangle+\left|111\right\rangle),\left|\psi^{2}\right\rangle=\frac{1}{ \sqrt{2}}(\left|001\right\rangle+\left|110\right\rangle)\\ \left|\psi^{3}\right\rangle=\frac{1}{\sqrt{2}}(\left|010\right\rangle+\left|1 01\right\rangle),\left|\psi^{4}\right\rangle=\frac{1}{\sqrt{2}}(\left|011 \right\rangle+\left|100\right\rangle)\\ \left|\psi^{5}\right\rangle=\frac{1}{\sqrt{2}}(\left|000\right\rangle-\left|1 11\right\rangle),\left|\psi^{6}\right\rangle=\frac{1}{\sqrt{2}}(\left|001 \right\rangle-\left|110\right\rangle)\\ \left|\psi^{7}\right\rangle=\frac{1}{\sqrt{2}}(\left|010\right\rangle-\left|1 01\right\rangle),\left|\psi^{8}\right\rangle=\frac{1}{\sqrt{2}}(\left|011 \right\rangle-\left|100\right\rangle)\end{array}  \tag{1}\]  
These GHZ states can be transformed into each other through unitary operations, such as  
\[ \begin{array}{l}\left|\psi^{2}\right\rangle=X\otimes X\otimes I\left|\psi^{ 1}\right\rangle,\quad\left|\psi^{3}\right\rangle=I\otimes X\otimes I\left| \psi^{1}\right\rangle\\ \left|\psi^{4}\right\rangle=X\otimes I\otimes I\left|\psi^{1}\right\rangle, \quad\left|\psi^{5}\right\rangle=Z\otimes I\otimes I\left|\psi^{1}\right\rangle \\ \left|\psi^{6}\right\rangle=ZX\otimes X\otimes I\left|\psi^{1}\right\rangle, \quad\left|\psi^{7}\right\rangle=Z\otimes X\otimes I\left|\psi^{1}\right\rangle \\ \left|\psi^{8}\right\rangle=ZX\otimes I\otimes I\left|\psi^{1}\right\rangle \end{array}  \tag{2}\]  
where \(X=\left( \begin{array}{cc}0&1\\ 1&0\end{array} \right)\), \(Z=\left( \begin{array}{cc}1&0\\ 0&-1\end{array} \right)\) are Pauli matrices, and \(I=\left( \begin{array}{cc}1&0\\ 0&1\end{array} \right)\) is the identity matrix. Assume that [PERSON] intends to transmit an arbitrary qubit \(\left|\chi\right\rangle_{a}\) to [PERSON], while [PERSON] intends to transmit \(\left|\chi\right\rangle_{b}\) to [PERSON] at the same time, \(\left|\chi\right\rangle_{a}\) and \(\left|\chi\right\rangle_{b}\) are given by  
\[ \begin{array}{l}\left|\chi\right\rangle_{a}=(\alpha|0\rangle+\beta|1\rangle )_{a}\\ \left|\chi\right\rangle_{b}=(\gamma|0\rangle+\delta|1\rangle)_{b}\end{array}  \tag{3}\]  
where \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) are complex probability amplitudes satisfying \(\left|\alpha\right|^{2}+\left|\beta\right|^{2}=1\), \(\left|\gamma\right|^{2}+\left|\delta\right|^{2}=1\). Without loss of generality, [PERSON] and [PERSON] previously share a six-particle GHZ-GHZ state, given by  
\[ \begin{array}{l}\left|\mathrm{C}\right\rangle\\ =\frac{1}{2}\left[\left(\left|00000\right\rangle+\left|00011\right\rangle+ \left|111000\right\rangle+\left|111111\right\rangle\right)_{A_{1}B_{1}A_{2}B_{2 }A_{3}B_{3}}\right]\end{array}  \tag{4}\]where the particles \(A_{1}\), \(A_{2}\) and \(A_{3}\) belong to [PERSON] and particles \(B_{1}\), \(B_{2}\) and \(B_{3}\) belong to [PERSON], as shown in Figure 1.  
The state of the initial eight-particle system \(|Y_{1}\rangle\) is given by  
\[ \begin{array}{l}|Y_{1}\rangle_{A_{1}B_{1}A_{2}B_{2}A_{3}B_{3}\Rightarrow}=|y ^{1}\rangle_{A_{1}B_{1}A_{2}}\otimes|y^{1}\rangle_{B_{2}A_{3}B_{3}}\otimes|x \rangle_{a}\otimes|x\rangle_{b}\\ =\frac{1}{2}\Big{[}(|000\rangle+|111\rangle)_{A_{1}B_{1}A_{2}}\otimes(|000 \rangle+|111\rangle)_{B_{2}A_{3}B_{3}}\otimes(\alpha|0\rangle+\beta|1\rangle) _{a}\otimes(\gamma|0\rangle+\delta|1\rangle)_{b}\end{array}  \tag{5}\]  
As shown in Figure 2, to achieve bidirectional qubit transmission, [PERSON] and [PERSON] perform Control-NOT (CNOT) gates and Hadamard gates in parallel, and the state of the entire system becomes  
\[