## 3 Bell State Exchange Parity Protocol and Intrinsic Autocompensation
### Autocompensating BSEP Protocol with Spatial Modes  
On the other hand, we can use the BSEP protocol with spatial modes (collinear or codirectional) with crosstalk between them in polarization-maintaining fibers (PANDA fibers, elliptical core fibers, and so on). Even the BSEP protocol with spatial modes should be valid in fibers with polarization crosstalk provided the final random polarization is the same in both modes. If the perturbation for the spatial modes on the first trip is \(P\) (a SU(2) matrix), then when the pulses return, it becomes \(P_{R}=P^{T}\), where \(T\) denotes the transpose. Therefore, for spatial modes, we have that [15; 21]  
\[P^{T}A_{s}P= \begin{pmatrix}\alpha&\beta\\ -\bar{\beta}&\bar{\alpha}\end{pmatrix}\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}\alpha&-\bar{\beta}\\ \beta&\bar{\alpha}\end{pmatrix} = \begin{pmatrix}0&1\\ -1&0\end{pmatrix} , \tag{20}\]  
where \(A_{s}\) is the spatial autocompensating matrix. As in the previous polarization-encoding case, the resulting transformation, represented by the matrix above, can easily be accounted for and removed at the end of the process. Thus, again, random perturbations are converted automatically into predictable transformations on the quantum states. On the other hand, note that in the spatial case, the propagation coefficients do not change sign when back propagating. For autocompensating with spatial modes we must distinguish between collinear and codirectional modes:  
* For collinear modes LP\({}_{11(\pm)}\), for example, HG modes, we use an arrangement of Dove prisms (DPs) rotated suitably, that is, DP\({}_{\theta}\), to implement the autocompensating matrix \(A_{s}\). The arrangement of DPs, as shown in Figure 4c, is as follows: First, we have DP\({}_{0}\), that is, a DP with its normal forming an angle equal to 0 with respect to direction \(Y\), producing a phase retarder \(\pi\), that is, implementing a \(Z\) transformation. Next, we have a DP\({}_{-\pi/4}\) implementing a -X transformation; therefore, the total transformation is \(-XZ=A_{S}\) as required. We must stress that cylindrical lens converters (CLCs) can also be used. A CLC is formed by two cylindrical lenses separated by an appropriate distance in such a way that introduces a selective Gouy phase in each spatial direction and therefore a relative phase between LP\({}_{11(\pm)}\) modes. For HG modes, the design is quite simple [38; 39]; however, for other type of modes, the design becomes more complex. * For the case of codirectional modes, counter-circulation of the modes plus a \(\pi\)-phase on one of them is enough to produce the desired autocompensating transformation \(A_{s}\), as indicated in Figure 4b; that is, it can be implemented by using two optical circulators and an isotropic \(\pi\)-phase shifter. For both spatial mode encodings, we disregard polarization, as commented above, in such a way that we can ignore the contribution of polarization coupling and only worry about spatial coupling. Under these conditions, the state arriving at Charlie will have the following form for codirectional and collinear modes, respectively,  
\[|L_{I,II}\rangle=\frac{1}{\sqrt{2}}(|1_{alp}1_{bII^{\prime}}\rangle\pm(|1_{alp }1_{bIp^{\prime}}\rangle),\quad|L_{X,Y}\rangle=\frac{1}{\sqrt{2}}(|1_{aXp}1_{bY ^{\prime}}\rangle\pm(|1_{aYp}1_{bXp^{\prime}}\rangle) \tag{21}\]  
where \(p\) and \(p^{\prime}\) denote random polarization states in each path \(a\) and \(b\). Note that we assume, in a fair approximation, the same polarization state for each pair of cores: \(p\) for [PERSON] and \(p^{\prime}\) for [PERSON] assume the same for the case of collinear modes, where this assumption  
Figure 4: Autocompensating loops for the BSEP protocol for the various encodings. (**a**) Polarization, (**b**) codirectional modes, (**c**) collinear modes. These loops are located at [PERSON] and [PERSON]’s stations and perform the required transformations. Then, the pulses return back to [PERSON].