## 4 Numerical Results and Comparisons
### Correlated Channels  
In Figures 7 and 8 we present the results for implementations of the SDC and LM05 for correlated channels compared to the asymmetrical BB84 in terms of their efficiencies against errors in the channel. We reiterate that we consider a particular model for correlated channels; _i.e._, specified by \(e_{1}=e_{2}=e_{m}\) and the error rates for SDC and LM05 in EM are given by \(q/2\) and \(q/4\) respectively [5].  
Figure 6: Efficiency for Asymmetrical BB84 (blue), LM05 (red) and SDC (green) against error rate \(q/2\) given independent channels in the infinite key limit  
Figure 7: Efficiency for Asymmetrical BB84 (blue), LM05 (red) and SDC (green) against error rate \(q/2\) for total qubit used as \(M=10^{4}\) for correlated channels specified by \(e_{1}=e_{2}=e_{m}\).  
In both the figures we see LM05 outperforming BB84 though in the second only marginally. This can be understood as due to going to infinite key limit where the two efficiencies should converge (binary entropic function for the bits to be discarded for both protocols in correlated channels is essentially given by the same form). This is illustrated in Figure 9 for increasing qubits used and further highlighted in Figure 10 where the black curve represents both the LM05 and BB84's key rate. However what is interesting is the fact that for finite number of qubits used, LM05 exceeds BB84 in terms of a higher tolerance towards errors clearly exemplified in Figure 7. This is almost unexpected as looking at the key rate formulas, one would imagine that the amount of bits to be discarded due to error correction and privacy amplification are essentially the same (relative to the key rates). However, a difference exists when considering the statistical fluctuation term. As LM05 allows for a larger number of bits to be salvaged in EM, thus a greater number of bits in CM (compared to BB84) may be used for error estimation in bounding Eve's information in the entropic relation. This allows for \(\mu(n_{e},k_{e})<\mu(n,k)\) implying  
\[h_{2}\left(\frac{q}{2}+\mu(n,k)\right)>h_{2}\left(\frac{q}{2}+\mu(n_{e},k_{e}) \right). \tag{27}\]  
Hence, for the same error in a channel, BB84 discards more bits than LM05, explaining LM05's higher error tolerance. This would be essentially the same reasoning for LM05's performance over SDC's in Figure 6.  
SDC's better performance when compared to BB84's in both figures and LM05's in Figure 7 is mainly due to the error correction term. In this particular model for correlated channels, the following inequality holds,  
\[h_{2}(q/2)\geq h_{4}(q/4), \tag{28}\]  
Figure 8: Efficiency for Asymmetrical BB84 (blue), LM05 (red) and SDC (green) against error rate \(q/2\) for total qubit used as \(M=10^{7}\) for correlated channels specified by \(e_{1}=e_{2}=e_{m}\).  
where less bits are discarded for error correction in SDC and equality holds only for \(q=0\). The infinite key regime of Figure 8 sees identical performance for both LM05 and BB84 while SDC exceeds both.  
Figure 10: Efficiency for Asymmetrical BB84, LM05 (both black) and SDC (green) against error rate \(q/2\) for correlated channels specified by \(e_{1}=e_{2}=e_{m}\) in the infinite key limit.  
Figure 9: Efficiency for Asymmetrical BB84 (blue), LM05 (red) and SDC (green) against total qubit used, \(M\) (for SDC, it is to be understood as total pairs used) for error rate \(0.01\) for correlated channels specified by \(e_{1}=e_{2}=e_{m}\). The horizontal dashed lines represents the infinite key regime.