# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 6 Some Final Considerations about Benchmarking, Decoherence Effects, and Fidelity
### QBER and a Brief Comparison with Other Similar QKD Protocols  
Thus, comparison between protocols is usually complicated because there are lots of elements to be performed. Moreover, some developments put more attention on certain variables to highlight the goodness of their approaches. In this subsection, we account for a review of similar BB84-like protocols in terms of some relative indicators for security. Despite the current development, the \(P_{E}\) is the outstanding feature because it measures the effective use of the sifting for [PERSON] (when she gets the correct outcome without detection), the most common comparative reference is the QBER. Thus, we show in the Appendix D, that the conditional QBER (relative to the useful outcomes when the basis of [PERSON] and [PERSON]\({}_{1}\)) becomes:  
\[P_{QBER_{nl}}=\frac{C_{0}(p_{0},K)}{P_{j}}K^{2}\sum_{k^{\prime}=0}^{1}\left[f _{k}(\omega_{1})f_{k^{\prime}}(\frac{\theta^{\prime}}{2})-(-1)^{k^{\prime}(k+ k^{\prime})}f_{k}(\omega_{1}+\frac{\Delta\theta}{2})f_{k\in\mathbb{I}^{k}}( \frac{\Delta\theta}{2})\right]^{2} \tag{32}\]  
Again, note the coefficient \(C_{0}(p_{0},K)\) is non-meaningful due \(P_{j}\), but instead, \(P_{QBER_{nl}}\) is proportional to \(K^{2}\). In addition, note that \(\theta^{\prime}\
eq\theta\), then \(\theta^{\prime}=\frac{\pi}{2}-\theta,\Delta\theta=\frac{\pi}{2}-2\theta\). Figure 9 shows the contour plots for \(P_{QBER_{nl}}\) as function of \(\omega\) and \(K\) corresponding to (a) \(k=0,\theta=0\); (b) \(k=0,\theta=\frac{\pi}{2}\); (c) \(k=1,\theta=0\); and (d) \(k=1,\theta=\frac{\pi}{2}\) remarking the region with \(P_{j}>0.9\). Each contour value of \(P_{QBER_{nl}}\) was coloured in agreement with the color bar besides. Inside the region shown with \(P_{j}\geq 0.9\) and \(0\leq K\leq 1\), the QBER drops in average around (a) 0.03, (b) 0.04, (c) 0.03, and (d) 0.02 relative to each figure. Despite, alternative elections with larger \(K\) and \(P_{j}\) values (as instance \(0.8\leq K\leq 20\) and \(P_{j}>0.95\)) raises those values to (a) 0.09, (b) 0.12, (c) 0.11, and (d) 0.07 respectively, near of the threshold for the QBER in the BB84 protocol [55], and under the security bound of 0.25 [56]. This fact is interesting because the method is configurable by selecting the region where \(P_{E}\) and \(P_{QBER_{nl}}\) are both satisfactory. Thus, if QBER is the main goal (the detection of eavesdropper more than its failure), then, regions with larger \(K\) values could be more practical. Still, some considerations should be analysed due to decoherence effects that could increase the QBER [57], thus reaching the security bound. It will be discussed in the next subsection. As it was stated in Section 5, the trend of BB84-like protocols has continued by proposing new approaches mainly based on this protocol. Thus, the BBM92 protocol [12] reports for individual attacks a theoretical QBER of \(\frac{1}{4}\) and a success probability of eavesdropping of \(\frac{1}{4}\) as the BB84 one. The SSP98 protocol [40] has reported theoretical QBER values of \(\frac{2}{3}\) and success probability of \(\frac{1}{3}\). Nevertheless, practical implementations report QBER's of 0.110 and 0.126% [58] for BB84 and SSP98 respectively. The SARG04 protocol [13] has reported QBER from 0.968% and 0.271% for single-photon and double photon pulses respectively. In newer protocols based on BB84 as MKP16 protocol [43] the QBER range from 0.56 until 0.25 depending on the initial number of qubits generated. Thus, the development of QKD protocols is not uniformly developed going first on the proposal, the QBER or success probability analysis, the attack type to be considered in the analysis. Each one could have different complexity, requiring further developments for its analysis through those several approaches highlighting their goodness.