## 2 BBM92 Protocol in 2D Subspace Fiber Encodings
### The Autocompensation Problem
First, additional circulation/paths are required in BBM92, which greatly increase the propagation distance and thus the attenuation. Secondly, polarization demultiplexing is also required; that is, the \(H\) and \(V\) modes are separated by a polarizing beam splitter at Charlie and then coupled to polarization-maintaining SMFs. Therefore, the way BBM92 works, it is not possible to devise a phase autocompensation system which, even assuming a maintained polarization, requires less propagation distance than three times the distance between Alice/Bob and Charlie. To briefly see how such a common path can achieve phase autocompensation, take the state (1) but consider the fact that now, due to the common path condition, \(\varphi_{a1}=\varphi_{a2}\) and \(\varphi_{b1}=\varphi_{b2}\). The resulting state is the original state (1), albeit a global phase, and thus the phase perturbation has been autocompensated. On the other hand, it is clear that general modal-coupling autocompensation is not possible; indeed, if the state starts from the end of [PERSON] and [PERSON], then, through a round trip, we could formally achieve autocompensation. Still, we need a Bell state shared between [PERSON] an [PERSON], who are in fact far apart. Consequently, the path between the source and [PERSON] and Bob would be left uncompensated. In short, polarization perturbations cannot be autocompensated by a single round trip, that is, twice the distance between [PERSON]/[PERSON] and Charlie. However, we achieve phase drift autocompensation via a triple trip, but we must assume that polarization is maintained. This is a strong limitation of the conventional SMFs used in optical fiber networks. Obviously, by using PANDA optical fibers, elliptical optical fibers, and so on, we could notably reduce cross polarization, and thus the triple trip topology could be used. Finally, we must stress that all the above autocompensation procedures can also be applied to spatial modes, that is, to codirectional modes as long as the cores are so far  
Figure 1: Autocompensating circuits for the BBM92 protocol with phase perturbations. We show here the general topology for polarization, collinear, and codirectional modes. Polarization and collinear modes encodings can be multiplexed/demultiplexed into codirectional modes \(1=H,2=V\) or \(1=X,2=Y\) by using polarizing beam-splitters or mode-sorting MZIs at the ends of the line; codirectional modes \(1=I,2=II\) are intrinsically de-multiplexed. Light travels from Charlie and ends at Alice/Bob. For simplicity, we only show half of the scheme’s layout; light circulation is analogous for Alice and for Bob. In both cases, light needs to travel three times the distance between Charlie and Alice/Bob. A photon in one mode travels the blue solid line and the other travels the red dashed line. The working principle of this topology is that both modes travel a common path assisted by Reconfigurable Directional Couplers (RDCs). The paths shown in the figure that are non-common can be made equal and error-free. apart that there is no spatial coupling but they still acquire different phases, or to collinear modes as long as there is no cross talking by using, for instance, elliptical optical fibers. Obviously, polarization must be preserved in all cases. Therefore, it is clear that the BBM92 protocol has very serious intrinsic limitations to be implemented in a plug and play way. Therefore, we will propose a new protocol also based on Bell states but this time with intrinsic advantages that make it applicable to plug and play implementations.