## 3 Bell State Exchange Parity Protocol and Intrinsic Autocompensation
### Autocompensating BSEP Protocol with Polarization Modes  
We must stress that this protocol, due to its two-way architecture, can inherently autocompensate for not only the accumulated propagation phases between \(a1\) and \(a2\) but also the coupling between them. In a general 2D subspace, perturbations can be removed. The two-way structure means that we now need to consider forward-propagation and back-propagation separately. The reasoning behind autocompensation is that, by means of a simple transformation between round trips, the random coefficients will end up cancelling themselves predictably.  
For the case of polarization modes, if the perturbation on the first trip is \(P\) (a SU(2) matrix), then when the pulses return, it becomes \(P_{R}=ZP^{\dagger}Z\), where \(T\) denotes the transpose and \(Z\) is the third Pauli matrix. Note that off-diagonal elements change sign, apart from being interchanged [15; 21], as the axis of propagation is modified when back-propagating, that is,  
\[ZP^{\dagger}ZA_{p}P=\begin{pmatrix}\alpha&-\beta\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\begin{pmatrix}\alpha&-\bar{\beta}\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}, \tag{17}\]where \(A_{p}\) is the polarization autocompensating matrix. This means that while before we had a transformation with unknown, random parameters, we now have a predictable transformation of the quantum state that can easily be reversed. This reversal is deterministic. All the random effects caused by the perturbation are removed automatically by just re-circulating the light back, without the need for active real-time monitoring or calibration. In Figure 4a, we show the device required to implement the above transformation, that is, a half-wave plate (HWP) rotated by \(\pi/4\) in a circular circuit. Recall that an \(\Omega\)-rotated HWP produces the following transformation on polarization modes (albeit containing a global phase which we omit for clarity):  
\[HWP_{\Omega}=\begin{pmatrix}\cos 2\Omega&\sin 2\Omega\\ \sin 2\Omega&-\cos 2\Omega\end{pmatrix}. \tag{18}\]  
The important conclusion is that by a single round trip, any perturbation can be autocompensated, even polarization-dependent absorption can be autocompensated as has been proven [36; 37]. The case of a phase drift can be cancelled by a round trip, unlike in the BBM92 protocol, where a triple trip is required and non cross-polarization was assumed.  
In order for this autocompensation technique to work, we need to delay pulses in one polarization with respect to the other [13; 21] as shown in Figure 2 with two pulses in each path. For instance, the pulse associated with \(|1_{jV}\rangle\) is delayed with respect to \(|1_{jH}\rangle\) by some fixed quantity \(\tau\), that is, \(|1_{jV}\rangle\rightarrow|1_{jV\tau}\rangle\). The delay is necessary because we need to separately phase modulate the perturbed \(|1_{jH}\rangle\) state with respect to \(|1_{jV}\rangle\). If not, the phase \(\theta\) we use for information encoding would be mixed in the random state coming out of the fiber. That would be equivalent to introducing a \(\theta-\)phase gate sandwiched between \(A_{p}\) and \(P\) in Equation (20). It is clear that the resulting transformation would not be \(A_{p}\), but another complicated expression involving \(\alpha\), \(\beta\), and \(\theta\) altogether; thus, autocompensation would not be achieved. What we actually have, because of the delay, is:  
\[\begin{split}|1_{jH}\rangle\to P_{R}A_{p}P|1_{jH}\rangle =A_{p}|1_{jH}\rangle,\\ |1_{jV}\rangle\to e^{i\theta}P_{R}A_{p}P|1_{jV\tau} \rangle=e^{i\theta}A_{p}|1_{jV\tau}\rangle.\end{split} \tag{19}\]  
Thus, correcting for the deterministic transform \(A_{p}\), we restore the original states sent by [PERSON]. Finally, note that the fact that autocompensation transformation interchanges the pulses; this guarantees that the pulses become synchronous again after traversing the delay when they return to Charlie, that is, \(|1_{jH}\rangle\rightarrow|1_{jH\tau}\rangle\).