# Research on Intraparticle to Interparticle Entanglement Swapping Protocols
## 3 Methods
### Entanglement Swapping Protocols
#### 3.1.3 Entanglement Swapping Protocols Based on the Cross-Kerr Medium  
The entanglement swapping protocol from spin-orbit intraparticle to spin-spin interparticle entanglement can be achieved using the cross-Kerr medium. We suppose that the intraparticle entangled states of photons 1 and 2 are as follows:  
\[|\varphi\rangle_{1}^{\prime}=\alpha|0\rangle_{1s}|0\rangle_{1o}+\beta|1 \rangle_{1s}|1\rangle_{1o}, \tag{8}\]  
and  
\[|\varphi\rangle_{2}^{\prime}=\alpha|0\rangle_{2s}|0\rangle_{2o}+\beta|1 \rangle_{2s}|1\rangle_{2o}, \tag{9}\]  
where \(|\alpha|^{2}+|\beta|^{2}=1,\left|\alpha\right|^{2}\) and \(\left|\beta\right|^{2}\) represent the probability of obtaining each part, respectively. To obtain the target entangled state (10), the protocol, as shown in Figure 4, is used to implement the entanglement swapping as is shown here:  
\[|\psi\rangle^{\prime}=\frac{1}{c}(|0\rangle_{1o}|0\rangle_{2o}+|1\rangle_{1 o}|1\rangle_{2o}), \tag{10}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Measurement Results** & **Corresponding Unitary Operators** \\ \hline \(|s^{+}\rangle\) & \(I\) \\ \(|s^{-}\rangle\) & \(\sigma_{z}\) \\ \(|k^{+}\rangle\) & \(\sigma_{x}\) \\ \(|k^{-}\rangle\) & \(i\sigma_{y}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measurement results and corresponding unitary operation operators. where \(\frac{1}{|c|^{2}}\) represents the probability of obtaining each part, \(\frac{2}{|c|^{2}}=1\). The joint state \(\ket{\varphi}_{3}^{\prime}\) composed of \(\ket{\varphi}_{1}^{\prime}\) and \(\ket{\varphi}_{2}^{\prime}\) is as shown here:  
\[\ket{\varphi}_{3}^{\prime}=\alpha^{2}\ket{00}_{1 so}\ket{00}_{2 so}+\alpha\beta( \ket{00}_{1 so}\ket{11}_{2 so}+\ket{11}_{1 so}\ket{00}_{2 so})+\beta^{2}\ket{11}_{1 so}\ket{11}_{2 so}, \tag{11}\]  
where \(\ket{\alpha}^{4}+\ket{\beta}^{4}+2\abs{\alpha}^{2}\ket{\beta}^{2}=1\). To achieve entanglement swapping, a half-wave plate placed at \(45^{\circ}\) is used to perform a spin-flip operation on the spin degrees of freedom of photons 1 and 2, and \(\ket{\varphi}_{3}^{\prime}\) becomes \(\ket{\varphi}_{4}^{\prime}\), which is represented by the following equation:  
\[\ket{\varphi}_{4}^{\prime}=\alpha^{2}\ket{10}_{1 so}\ket{10}_{2 so}+\alpha\beta( \ket{10}_{1 so}\ket{01}_{2 so}+\ket{01}_{1 so}\ket{10}_{2 so})+\beta^{2}\ket{01}_{1 so}\ket{01}_{2 so}. \tag{12}\]  
Next, photons 1 and 2 pass through the QND [39], and \(\ket{\varphi}_{4}^{\prime}\) becomes \(\ket{\varphi}_{5}^{\prime}\), as is represented by the following equation:  
\[\ket{\varphi}_{5}^{\prime}=\left(\alpha^{2}\ket{10}_{1 so}\ket{10}_{2 so}+\beta^ {2}\ket{01}_{1 so}\ket{01}_{2 so}\right)\ket{he^{i\theta}}+\alpha\beta\ket{h} \left(\ket{10}_{1 so}\ket{01}_{2 so}+\ket{01}_{1 so}\ket{10}_{2 so}\right|e^{2i \theta}\right), \tag{13}\]  
where \(\ket{h}\) represents incident light in QND, and \(\theta\) represents the phase shift. The corresponding unitary operations are performed on the QND to make \(\theta=\pi\). Photons pass through the QND. If the measurement result of phase shift is 0 or 2\(\pi\), which indicates a difference in polarization states between two photons [37], \(\ket{\varphi}_{5}^{\prime}\) collapses to \(\ket{\varphi}_{6}^{\prime}\), as shown in the following equation:  
\[\ket{\varphi}_{6}^{\prime}=\sqrt{(1-\alpha^{4}-\beta^{4})}(\ket{10}_{1 so}\ket {01}_{2 so}+\ket{01}_{1 so}\ket{10}_{2 so}). \tag{14}\]  
Next, photons 1 and 2 pass through a half-wave plate (placed at 22.5\({}^{\circ}\)) that can achieve a Hadamard operation on the spin degree of freedom of the photons.