## 2 BBM92 Protocol in 2D Subspace Fiber Encodings
### The Autocompensation Problem  
Let us consider the usual polarization encoding in the BBM92 protocol. By direct inspection, it can be seen that the standard topology, Alice \(\leftarrow\) Charlie \(\rightarrow\) Bob, of this protocol does not allow the passive autocompensation of cross polarization; not even a simple phase drift can be autocompensated. Indeed, since the quantum source is located at Charlie, the usual round trip technique cannot be used. The only possibility of achieving passive autocompensation would be to change the standard topology. To make this possibility clear, we start by considering the simplified situation where only phase drift exists. This corresponds to the case where the two polarization modes have different propagation constants due to birefringence but without polarization coupling. We will show how to achieve a plug and play topology for autocompensation of phase drifts in [PERSON]'s and [PERSON]'s paths. We insist that polarization modal coupling cannot be autocompensated. In formal terms, the phase drift can be obtained by setting \(\theta=\omega=0\) in Equation (8); therefore, the perturbation becomes simply a phase gate, that is,  
\[P_{phase}= \begin{pmatrix}e^{-i\varphi/2}&0\\ 0&e^{i\varphi/2}\end{pmatrix} . \tag{9}\]  
Now, if the perturbation takes the form of Equation (9), the error probabilities are much simpler. Indeed, we find that \(p_{err}^{Z}=0\), which makes sense, as phase drift cannot make photons jump between H and V polarizations (and equivalently for the other encodings). The effect of phase drift accumulates relative phases; a bit encoded as \(|1_{H}\rangle\) cannot become \(|1_{V}\rangle\) under the influence of phase drift. This is not the case for a bit encoded as \(|1_{H}\rangle+|1_{V}\rangle\), which may become the opposite encoded bit if a phase \(\pi\) happens to accumulate during propagation, thus producing the state \(|1_{H}\rangle-|1_{V}\rangle\) (normalization factors aside). Indeed, for the \(X\) basis, we obtain  
\[p_{err}^{X}=\frac{1}{2}[1-\cos((\varphi_{a2}-\varphi_{a1})-(\varphi_{b2}- \varphi_{b1}))]=\frac{1}{2}[1-\cos(\varphi_{a}-\varphi_{b})]. \tag{10}\]  
that is, states in the \(X\) basis are sensible to phase perturbations. Therefore, the entangled state (1) becomes  
\[|\Psi^{-}\rangle=\frac{1}{\sqrt{2}}(e^{i(\varphi_{a1}+\varphi_{b2})}|1_{a1}1_{ b2}\rangle+e^{(i(\varphi_{a2}+\varphi_{b1})}|1_{a2}1_{b1}\rangle),\]  
thus accumulating a relative phase \((\varphi_{a2}-\varphi_{a1})-(\varphi_{b2}-\varphi_{b1})=\varphi_{a}-\varphi_ {b}\). One can see here that this relative phase does not affect measurements in the Z basis, as the correlations [PERSON] and [PERSON] would measure would remain unchanged. When postulating probability distributions (PDFs) for these random variables, we shall consider the phase difference \(\varphi_{a}=\varphi_{a2}-\varphi_{a1}\) altogether (analogously for \(\varphi_{b}\)). That is, the PDF is a statistical distribution of the phase difference, and not of the individual accumulated phases. This will become clearer in the following sections. We must stress that we maintain the general subscripts \(a1,a2,b1,b2\), although for polarization they should be \(aH,aV,bH,bV\), whereas the same treatment can be applied to collinear \((X,\gamma)\) codirectional \((I,II)\) modes, as mentioned above. The simplest mechanism in order to passively remove the phase perturbation is to make the modes travel the same path. This way, we obtain a common phase that can be removed. However, in this case, it is not enough to make light travel back and forth, since the state is generated at Charlie but is detected at Alice and Bob. Therefore a plug and play based on a single round trip as used, for instance, in [14; 21], is not valid for the BBM92 QKD protocol. It requires as a minimum a triple trip (forth-back-forth). The layout of the fiber-optic circuit implementing autocompensation is shown in Figure 1.  
The figure presents two main differences with respect to the single-photon phase auto-compensation technique shown in [14].