## 4 Simulation Results and Discussion
In particular, it is known that effective error correction and privacy amplification protocols exist for the BB84 protocol when the QBER is below 11%, and this threshold is believed to be acceptable for all similar QKD protocols [24; 25]. MDI-QKD, as we have formulated it, is not identical to BB84. However, it is similar in that, from the point of view of [PERSON], MDI-QKD is functionally equivalent to BB84 if she conceptualizes [PERSON] as an agent of [PERSON]. Accordingly, we expect the threshold QBER for MDI-QKD to be no less than 11%, which is significantly above the QBER we obtained in our simulation of realistic source non-idealities. The practical implementation of MDI-QKD depends on limiting bit errors due to incomplete HOM interference. By simulating the MDI-QKD protocol using photon source tolerances from a well-characterized two-photon interference study, we demonstrate both good agreement with experiment and the ability to relate the achievable visibility to fluctuations in specific photon properties. From the plots in Figure 4, it is clear that the QBER is sensitive to both photon wavelength and pulse  
Figure 4: Dependence of the QBER on photon source tolerances for a single variable. The QBER values are averaged over 100 trials of \(N_{raw}=2500\) with uncertainty dominated by shot noise. Error bars represent three times the standard deviation of the mean in either direction. timing. Modest increases in the trial-to-trial fluctuations of photon wavelengths and arrival times lead to significant increases in the QBER, while suppressing either of these fluctuations results in a factor of two improvement. In contrast, the QBER is relatively insensitive to changes in the bandwidth standard deviation and polarization angle standard deviation, even if these fluctuations are suppressed entirely. The sensitivity of the QBER to wavelength and timing fluctuations points to the need for design tradeoffs to optimize system performance. In the experiment of [17], wavelength fluctuations are limited by narrow filters that block out-of-tolerance photons from reaching the detectors. While this enhances the HOM visibility, it also reduces the detection rate. As QKD performance is measured on the basis of the rate at which secret key can be generated, there is a point of diminishing returns when the QBER is improved at the expense of transmission rate. Similar reasoning applies to photon synchronization if improvements in the timing of pulse generation come at the expense of a reduced probability of generating a photon. A reduction in the detection rate could also occur with tighter polarization angle and pulse bandwidth control without improvement in the QBER, thus reducing overall QKD system performance. Even though the QBER is not particularly sensitive to bandwidth fluctuations in the region of parameter space under consideration, the sensitivity to pulse synchronization is closely related to the nominal bandwidth. As the bandwidth is inversely proportional to the temporal width of the pulse, larger bandwidth pulses are shorter in duration, thus requiring tighter synchronization tolerances to ensure large wave function overlap, reliable HOM interference, and a low QBER. We also note that the use of very narrow wavelength filters has the potential to complicate the analysis by altering the shape of the photon wave function.