# Research on Intraparticle to Interparticle Entanglement Swapping Protocols
## 3 Methods
### Entanglement Swapping Protocols
#### 3.1.4 Entanglement Swapping Protocols Based on Linear Optical Elements  
We assume that the intraparticle entangled states of photons 1 and 2 are (8) and (9). To obtain the desired entangled state (10), the protocol, as shown in Figure 5, is used to implement entanglement swapping. The joint state \(|\varphi\rangle_{3}^{\prime\prime}\) composed of (8) and (9) is as follows:  
\[|\varphi\rangle_{3}^{\prime\prime}=\alpha^{2}|00\rangle_{1 so}|00\rangle_{2 so} +\alpha\beta(|00\rangle_{1 so}|11\rangle_{2 so}+|11\rangle_{1 so}|00\rangle_{2 so })+\beta^{2}|11\rangle_{1 so}|11\rangle_{2 so}. \tag{21}\]  
The spin-flip operation is performed on the spin degrees of freedom of photons 1 and 2 by the half-wave plates placed at \(45^{\circ}\), and therefore (21) becomes (22), as is shown by the following equation:  
\[|\varphi\rangle_{4}^{\prime\prime}=\alpha\beta(|1\rangle_{1s}|0\rangle_{2s} |0\rangle_{1o}|1\rangle_{2o}+|0\rangle_{1s}|1\rangle_{2s}|1\rangle_{1o}|0 \rangle_{2o})\]  
\[\alpha^{2}|1\rangle_{1s}|1\rangle_{2s}|0\rangle_{1o}|0\rangle_{2 so}+\beta^{2} |0\rangle_{1s}|0\rangle_{2s}|1\rangle_{1o}|1\rangle_{2o}\]  
Figure 5: The experimental schematic of entanglement swapping based on linear optical elements. R\({}_{45}\) represents a half-wave plate placed at \(45^{\circ}\), R\({}_{22.5}\) represents a half-wave plate placed at \(22.5^{\circ}\), PBS represents the polarization beam splitter, and D\({}_{3(i=1,2,3,4)}\) represents detectors. In the above equation, the first and second terms will cause the transmission and reflection spatial modes of PBS1 to output photons. The third and fourth terms can only cause one spatial mode (transmission or reflection) of PBS1 to output photons. Thus, if photons output from both the transmission and reflection spatial modes of PBS1, \(|\varphi\rangle_{4}^{\prime\prime}\) becomes \(|\varphi\rangle_{5}^{\prime\prime}\). \[|\varphi\rangle_{5}^{\prime\prime}=\sqrt{(1-\alpha^{4}-\beta^{4})}(|1\rangle_{ 1s}|0\rangle_{2s}|0\rangle_{1o}|1\rangle_{2o}+|0\rangle_{1s}|1\rangle_{2s}|1 \rangle_{1o}|0\rangle_{2o}). \tag{23}\]  
Next, photons pass through a half-wave plate placed at \(22.5^{\circ}\), which can achieve a Hadamard operation for the spin degree of freedom of photons, and \(|\varphi\rangle_{5}^{\prime\prime}\) becomes \(|\varphi\rangle_{6}^{\prime\prime}\), which is represented by the following equation:  
\[|\varphi\rangle_{6}^{\prime\prime}=\frac{\sqrt{(1-\alpha^{4}-\beta^{4})}}{2}[( |01\rangle_{1o2o}+|10\rangle_{1o2o})(|00\rangle_{1s2s}-|11\rangle_{1s2s})+(|01 \rangle_{1o2o}-|10\rangle_{1o2o})(|01\rangle_{1s2s}-|10\rangle_{1s2s})]. \tag{24}\]  
In Equations (23) and (24), the subscripts \(1s2s\) and \(1o2o\) represent the spin and orbital angular momentum degrees of freedom of photons 1 and 2, respectively. Next, photons pass through a PBS, and four photon detectors are used to determine the results. If one of the four detectors rings twice, the entangled state \(|\varphi\rangle_{6}^{\prime\prime}\) will collapse, as is represented by the following equation:  
\[|\varphi\rangle_{7}^{\prime\prime}=\frac{\sqrt{(1-\alpha^{4}-\beta^{4})}}{2}(| 01\rangle_{1o2o}+|10\rangle_{1o2o}). \tag{25}\]  
The unitary operator \(\sigma_{x}\) is performed on the orbital angular momentum degree of freedom of photon 2; the desired entangled state is obtained as the following:  
\[|\varphi\rangle_{8}^{\prime\prime}=\frac{\sqrt{(1-\alpha^{4}-\beta^{4})}}{2}( |00\rangle_{1o2o}+|11\rangle_{1o2o}). \tag{26}\]  
If detectors \(D_{1},D_{2}\) or \(D_{3}\), \(D_{4}\) ring, the entangled state \(|\varphi\rangle_{6}^{\prime\prime}\) will collapse, as is represented by the following equation:  
\[|\varphi\rangle_{9}^{\prime\prime}=\frac{\sqrt{(1-\alpha^{4}-\beta^{4})}}{2}( |01\rangle_{1o2o}-|10\rangle_{1o2o}). \tag{27}\]  
The desired entangled state will be obtained by performing the unitary operation \(i\sigma_{y}\) on the orbital angular momentum degree of freedom of photon 2.