## 2 BBM92 Protocol in 2D Subspace Fiber Encodings
### Bell States for BBM92 Protocol  
Consider the BBM92 protocol [18], where [PERSON] and Bob are separated by some distance and each receive a part of an entangled photonic state. The entangled state comes from a station located in the middle of them. We may call it Charlie, in analogy with MDI-QKD schemes. At Charlie, there is a source emitting singlet states of the form  
\[|\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|1_{a1}1_{b2}\rangle-|1_{a2}1_{b1}\rangle), \tag{1}\]  
This is a maximally entangled Bell state, and moreover, it is a basis-invariant state, as shown below. In principle, we could also use the Bell state \(|\phi^{+}\rangle=2^{-1/2}(|1_{a1}1_{b1}\rangle+|1_{a2}1_{b2}\rangle)\) because \(|\phi^{+}\rangle\) is also basis-invariant. Nevertheless, note that the states \(|\psi^{+}\rangle=2^{-1/2}(|1_{a1}1_{b2}\rangle+|1_{a2}1_{b1}\rangle)\) and \(|\phi^{-}\rangle=(|1_{a1}1_{b1}\rangle-|1_{a2}1_{b2}\rangle)\) are not basis-invariant, and therefore are less robust than the above states. Nevertheless, \(|\phi^{+}\rangle\) and \(|\phi^{-}\rangle\) cannot be distinguished by linear optics, which will be important for the protocol we propose later. In short, \(|\psi^{-}\rangle\) is the optimal choice. The notation used summarizes any of the encodings under consideration. For instance, \(|1_{a1}\rangle\) may be equivalent to \(|1_{aH}\rangle\), identifying the state of an H-polarized photon on Alice's channel, while \(|1_{a2}\rangle\) will do so for a V-polarized photon \(|1_{aV}\rangle\). Furthermore, the subscripts 1 and 2 identify the cores of a two-core fiber \(I\) and \(II\), that is, \(|1_{aI}\rangle\) and \(|1_{aII}\rangle\), as in the usual dual-rail encoding (codirectional modes). Moreover, \(|1_{a1}\rangle\) may refer to LP modes in FMFs, which describe a very wide family of modal solutions in optical fibers, that is, \(E_{lp}(\rho,\varphi)=F_{p}(\rho)e^{il\varphi}\), which in turn transport orbital angular momentum \(l\), that is, OAM modes. The radial part determines the type of mode, that is, Hermite-Gaussian modes, Laguerre-Gaussian modes, and so on. For the sake of expositional convenience and without loss of generality, we choose the so-called LP\({}_{11(\pm)}\) modes. These modes can be expressed as follows: LP\({}_{11(\pm)}=(1/\sqrt{2})(E_{10}\pm E_{-10})\) which present a zero along the \(X\) and \(Y\) directions, respectively. Accordingly, we can have single-photon states excited in these modes; that is, \(|1_{a1}\rangle\) can refer to \(|1_{a+}\rangle\equiv|1_{aX}\rangle\), while \(|1_{a2}\rangle\) refers to \(|1_{a-}\rangle\equiv|1_{aY}\rangle\). Finally, we could particularize these modes for particular \(F(\rho)\) functions. For example, if the optical fiber admits the radial solution \(F(\rho)=F_{\varphi}\rho e^{-\rho^{2}/w_{o}^{2}}\) (parabolic approximation of a graded index optical fiber), where \(w_{o}\) is a constant and \(F_{o}\) is a normalization constant, the above LP modes become Hermite-Gaussian modes H\({}_{10}\) and H\({}_{10}\), where the subindices indicate the number of zeros along the \(X\) and \(Y\) directions. We will use the notation \(X\) and \(Y\) for LP\({}_{11(\pm)}\) modes. Obviously, we could choose other linear combinations, or even the OAM modes \(E_{\pm 11}(\rho,\varphi)\), but no new result would be obtained in our study. We must indicate that we will restrict ourselves to 2D subspaces. For polarization, this is the natural thing to do, but for collinear codirectional modes, we will need to assume that polarization is common to both modes. In terms of actual implementations, the condition of a common polarization is realistic, albeit restrictive, for the case of codirectional modes if we consider, for instance, (multicore) PANDA fibers [25] with reduced polarization mode couplings. For the case of collinear modes, a regular FMF could be used. In this case, it is much more reasonable to assume a common polarization, as the modes travel the same core of the same fiber, although elliptical optical fibers could be used to avoid modal coupling [14].