# Research on Intraparticle to Interparticle Entanglement Swapping Protocols
## 3 Methods
### Entanglement Swapping Protocols
#### 3.1.2 Entanglement Swapping Protocols Based on Bell State Joint Measurement  
This section presents an entanglement swapping protocol from spin-orbit intraparticle to spin-spin interparticle entanglement based on Bell state joint measurement. We suppose that the spin-orbit intraparticle entangled states are  
\[\ket{\varphi}_{1}=\frac{1}{\sqrt{2}}(\ket{0}_{1s}\ket{0}_{1o}+\ket{1}_{1s} \ket{1}_{1o}), \tag{1}\]  
and  
\[\ket{\varphi}_{2}=\frac{1}{\sqrt{2}}(\ket{0}_{2s}\ket{0}_{2o}+\ket{1}_{2s} \ket{1}_{2o}), \tag{2}\]  
Figure 3: Entanglement swapping principle. SAM represents the spin angular momentum, and OAM represents the orbital angular momentum. One circle represents one degree of freedom, and a black dot represents a particle.  
where \(1s,1o\) and \(2s,2o\) represent the spin and orbital angular momentum degrees of freedom of photons 1 and 2, respectively. The target entangled state between two photons is as follows:  
\[|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{1o}|0\rangle_{2o}+|1\rangle_{1o}|1 \rangle_{2o}), \tag{3}\]  
and the joint state of Equations (1) and (2) can be written as shown here:  
\[|\varphi\rangle_{3}=\frac{1}{2}\bigg{(}\begin{array}{c}|0\rangle_{1s}|0 \rangle_{1o}|0\rangle_{2o}|0\rangle_{2o}+|0\rangle_{1s}|0\rangle_{1o}|1 \rangle_{2s}|1\rangle_{2o}\\ +|1\rangle_{1s}|1\rangle_{1o}|0\rangle_{2s}|0\rangle_{2o}+|1\rangle_{1s}|1 \rangle_{1o}|1\rangle_{2s}|1\rangle_{2o}\end{array}\bigg{)}. \tag{4}\]  
United measurement is executed on the spin degree of freedom of two photons by using the Bell basis. The Bell basis is shown in the following equations:  
\[\begin{array}{c}|\mathrm{g}^{\pm}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{1s}| 0\rangle_{2s}\pm|1\rangle_{1s}|1\rangle_{2s})\\ |k^{\pm}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{1s}|1\rangle_{2s}\pm|1\rangle_{1 s}|0\rangle_{2s})\end{array}. \tag{5}\]  
Equation (4) can be rewritten as (6) using the Bell basis, as follows:  
\[\begin{array}{c}|\varphi\rangle_{4}=\frac{1}{2\sqrt{2}}\bigg{[}\begin{array} []{c}|\mathrm{g}^{+}\rangle(|1\rangle_{1o}|1\rangle_{2o}+|0\rangle_{1o}|0 \rangle_{2o})+|\mathrm{g}^{-}\rangle(|1\rangle_{1o}|1\rangle_{2o}-|0\rangle_{ 1o}|0\rangle_{2o})\\ +|k^{+}\rangle(|0\rangle_{1o}|1\rangle_{2o}+|1\rangle_{1o}|0\rangle_{2o})+|k^ {-}\rangle(|0\rangle_{1o}|1\rangle_{2o}-|1\rangle_{1o}|0\rangle_{2o})\end{array} \bigg{]}.\end{array} \tag{6}\]  
According to (6), the desired entanglement swapping state \(|\psi\rangle\) can be directly obtained when the joint measurement result is \(|g^{+}\rangle\). If the joint measurement result is otherwise, a corresponding unitary operation is carried out on the orbital angular momentum degree of freedom of photon 2 to obtain the necessary entangled state. The measurement results and corresponding unitary operation operators are shown in Table 1.  
The expression of these unitary operators is as follows:  
\[I=\begin{pmatrix}1&0\\ 0&1\end{pmatrix},\sigma_{z}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix},\;\sigma_{x}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix},\;i\sigma_{y}=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}. \tag{7}\]