## 2 BBM92 Protocol in 2D Subspace Fiber Encodings
### Perturbations in Quantum Light States  
Fibers are imperfect and undergo external perturbations, giving rise to perturbations affecting quantum light states. This means that we will have random relative phases, unpredictable couplings between modes, and even rotations. To model such effects, we assume that we can represent the perturbations as two general SU(2) transformations. These matrices have random parameters, and each one acts locally on [PERSON]'s and [PERSON]'s photonic mode operators. Specifically, we can write the following 4 \(\times\) 4 matrix, acting on the absorption operator vector \(\{\hat{a}_{a1},\hat{a}_{a2},\hat{a}_{b1},\hat{a}_{b2}\}^{T}\)  
\[P_{ab}= \begin{pmatrix}\alpha_{a}&-\bar{\beta}_{a}&0&0\\ \beta_{a}&\bar{\alpha}_{a}&0&0\\ 0&0&\alpha_{b}&-\bar{\beta}_{b}\\ 0&0&\beta_{b}&\bar{\alpha}_{b}\end{pmatrix} . \tag{3}\]  
where the bar stands for conjugation. Therefore, we obtain the following transformation of absorption operators  
\[\hat{a}_{1}^{\dagger}\rightarrow\alpha_{j}\hat{a}_{1}^{\dagger}+\beta_{j} \hat{a}_{2}^{\dagger},\;\;\;\hat{a}_{2}^{\dagger}\rightarrow-\bar{\beta}_{j} \hat{a}_{1}^{\dagger}+\bar{\alpha}_{j}\hat{a}_{2}^{\dagger}, \tag{4}\]  
where, again, \(j=a,b\), as the perturbation coefficients are different for [PERSON] and [PERSON]. Moreover, by recalling the fundamental relationships \(\hat{a}_{1}^{\dagger}|0\rangle=|1_{1}\rangle\), \(\hat{a}_{2}^{\dagger}|0\rangle=|1_{2}\rangle\), and by taking into account Equation (4), we see how the singlet state (1) is modified by the action of the perturbations, becoming  
\[ \begin{split}|\psi^{-}\rangle\to 2^{-1/2}\{(\bar{\beta}_{a} \alpha_{b}-\alpha_{a}\bar{\beta}_{b})|1_{a1}1_{b1}\rangle+(\alpha_{a}\bar{ \alpha_{b}}+\bar{\beta}_{a}\beta_{b})|1_{a1}1_{b2}\rangle\\ -(\bar{\alpha}_{a}\alpha_{b}+\beta_{a}\bar{\beta}_{b})|1_{a2}1_{ b1}\rangle+(\beta_{a}\bar{\alpha}_{b}-\bar{\alpha}_{a}\beta_{b})|1_{a2}1_{ b2}\rangle\}.\end{split}  \tag{5}\]  
We can already read from here how the perfect (anti-)correlations of the singlet state are disturbed by the perturbations. In fact, whenever [PERSON] and [PERSON] both happen to measure in the \(Z\) basis, errors will occur whenever their measurement outcome is \(|1_{a1}1_{b1}\rangle\) or \(|1_{a2}1_{b2}\rangle\), which happens with probability  
\[p_{err}^{Z}=1/2(|\bar{\beta}_{a}\alpha_{b}-\alpha_{a}\bar{\beta}_{b}|^{2}+| \beta_{a}\bar{\alpha}_{b}-\bar{\alpha}_{a}\beta_{b}|^{2}). \tag{6}\]Given that the ideal state is a singlet, either [PERSON] or [PERSON] needs to apply a bit-flip to their measured output. If [PERSON] and [PERSON]'s results are correlated rather than anti-correlated, the bit value will be wrongfully assigned and they will share a string with erroneous bits in such cases. After [PERSON] or [PERSON] broadcasts their choice of basis, each one of them knows that the other one obtained the opposite state. Thus, if everything works fine and [PERSON] measures \(|1_{a1}\rangle\), then [PERSON] must have measured \(|1_{b2}\rangle\). [PERSON] writes down a 0 as her bit. [PERSON] writes down 1 and bit-flips to 0 (alternatively, it can be [PERSON] who does this). They share the same bit: 0. However, if errors are present, [PERSON] may measure \(|1_{a1}\rangle\) and at the same time [PERSON] may measure \(|1_{b1}\rangle\). [PERSON] would assign bit 0 and [PERSON] wrotes down 0 and bit-flips to 1, so they no longer share the same string. Table 1 summarizes such situations, also for the X basis, in the polarization encoding for simplicity. Something similar takes place for measurements in the \(X\) basis. First of all, take into account that for this measurement, [PERSON] and [PERSON] need to apply a Hadamard transform first, that is, relationship (2), to their qubits, respectively. Secondly, the state (1) is invariant under a change of basis, so in the \(X\) basis, [PERSON] and [PERSON] also expect perfectly anti-correlated results upon measurement. However, the action of the perturbation will prevent this from happening a fraction of the time.