In this section, we present numerical results illustrating the impact of device imperfections at Bob’s detection system on the estimated channel parameters and subsequently on the estimated secret key rate. In Fig. 2 (a) we present the evolution of the estimated channel transmission, expressed in (20), as a function of the transmission parameter of Bob’s beam-splitter, \(\eta _{\text{B}}\) in Fig. 1. From the results in Fig. 2 (a) we can see that when the system is balanced, i.e. \({\S }\eta _{\text{B}} = 0.5\), the transmission estimated by Bob will coincide with the real transmission. However, as it further imbalances, the estimated transmission will follow the curve dictated by \(\sqrt{\eta _{\text{B}}(1-\eta _{\text{B}})}\). Note that the first term in (16) does not have a major contribution in (20), due to the low power of the quantum signal. The second term in (16), due to it being a purely DC contribution and DC being filtered out by the bandpass filter \(h_{\text{BP}}(t)\), also does not have a major contribution to (20).
As stated previously, Bob’s shot noise estimate, Σ, will have contributions from both his laser’s shot noise and RIN. In Fig. 2 (b), we show the dependency of the noise parameters in (21) with Bob’s beam-splitter transmission coefficient. From the results in Fig. 2 (b) we can see that the shot noise contribution remains unchanged with the imbalances, which is to be expected, as the shot noise term in (21) is not dependent on \(\eta _{\text{B}}\) when \({\eta _{\text{d}_{1}}=\eta _{\text{d}_{2}}}\). Meanwhile, the RIN’s contribution rises sharply and rapidly becomes the dominant factor to the global noise at Bob’s detection output. The result of the combined shot noise and RIN is an overestimation of the shot noise, which will influence the estimation of the excess noise in relation to it, which can be seen in Fig. 2 (c).
In Fig. 2 (c), we present the evolution of the estimated excess of noise given by (22) in SNU, as a function of Bob’s beam-splitter imbalance. The excess noise presented here is due only to Alice’s RIN, shot noise and the shot noise of the interference between Alice’s modulated signal and Bob’s LO, corresponding to the second, fourth and sixth terms of (18), respectively. In Fig. 2 (c), we can see that, for the different values of RIN, the excess noise will always follow roughly the same behaviour. The estimated excess noise is maximum when the system is balanced and decreases symmetrically around \(\eta _{\text{B}}=0.5\) as it imbalances. This is due to Bob’s estimation for the shot noise, Σ, increasing sharply as the system becomes unbalanced, as shown in Fig. 2 (b). The higher the RIN, the more pronounced the excess noise underestimation will be. The decrease of the estimated transmission will also have an impact on the estimated excess noise, as the estimated transmission decreases with the increasing imbalance of \(\eta _{\text{B}}\), the estimated excess noise would also increase. However the effect of the increase of Σ̃ will be the dominant factor.
The secret key rate can then be estimated for the estimated values of transmission and excess noise following (23), yielding the results presented in Fig. 2 (d). For the lowest value of RIN the estimated secret key rate will decrease as the \(\eta _{\text{B}}\) deviates from 0.5, with this effect being dictated by the decreasing estimated transmission observed in Fig. 2 (a). This results in some lost system performance, as usable bits will be discarded. For the other two values of RIN, the underestimation of the excess noise observed in Fig. 2 (c) will cause an overestimation of the secure key rate, this poses a security risk as Alice and Bob will distill bits for the key at a rate higher than the channel parameters would allow for a secure key.
In Fig. 3 (a), we present the evolution of the estimated channel transmission, defined in (20), as a function of the difference between the quantum efficiencies of Bob’s photodiodes, identified by \(\eta _{\text{d1}}\) and \(\eta _{\text{d2}}\) in Fig. 1, for three different values of \(\eta _{\text{B}}\). From the results in Fig. 3 (a) we can see that the estimated transmission follows the curve dictated by \({\eta _{\text{d1}}-\eta _{\text{d2}}}\), with the different values of \(\eta _{\text{B}}\) causing a small vertical shift to the curve. When \({\eta _{\text{d1}}-\eta _{\text{d2}}>0}\) the transmission will tend to be overestimated, while when \({\eta _{\text{d1}}-\eta _{\text{d2}}<0}\) it will tend to be underestimated. Furthermore, we can see that equal deviations of \({\eta _{\text{B}}}\) in either direction, i.e. \({\eta _{\text{B}}<0.5}\) and \({\eta _{\text{B}}>0.5}\), will result in the same deviation of the estimated channel transmission.
In Fig. 3 (b), we show the dependency of Σ̃, expressed in (21), with the difference between the quantum efficiencies of Bob’s photodiodes. We see from Fig. 3 (b) that, when \({\eta _{\text{B}}=0.5}\), Bob’s estimated shot noise has a minimum value, corresponding to the true shot noise, at \({\eta _{\text{d1}}=\eta _{\text{d2}}}\), and rises as the quantum efficiencies deviate, as the RIN contribution becomes more and more pronounced. Moreover, in Fig. 3 (b), we see that when the value of \(\eta _{\text{B}}\) deviates from 0.5, the point at which the value of Bob’s estimated shot noise is minimum deviates to negative values of \({\eta _{\text{d1}}-\eta _{\text{d2}}}\) when \({\eta _{\text{B}}<0.5}\) and to positive values of \({\eta _{\text{d1}}-\eta _{\text{d2}}}\) when \({\eta _{\text{B}}>0.5}\). This hints at the fact that imbalances in Bob’s beam-splitter may be compensated by tuning the relative values of \(\eta _{\text{d1}}\) and \(\eta _{\text{d2}}\) and vice versa. Additionally, the value of Bob’s estimated shot noise at this minimum point will be slightly below the value observed with the balanced system when \({\eta _{\text{B}}<0.5}\) and, conversely, slightly above that value when \({\eta _{\text{B}}>0.5}\). This asymmetry is due to the average value of the quantum efficiencies of the photodiodes being lower when \({\eta _{\text{d1}}<\eta _{\text{d2}}}\) and higher when \({\eta _{\text{d1}}>\eta _{\text{d2}}}\), causing the second term of (21), which corresponds to Bob’s laser’s shot noise, to increase as the value of \({\eta _{\text{d1}}-\eta _{\text{d2}}}\) increases.
In Fig. 3 (c), we present the evolution of the estimated excess noise, given by (22), in SNU, as a function of the difference between the quantum efficiencies of Bob’s photodiodes, for three different values of \(\eta _{\text{B}}\). We can see from Fig. 3 (c) that all the estimated excess noise curves tend to the same value at \({\eta _{\text{d1}}=\eta _{\text{d2}}}\), with the excess noise being overestimated when \({\eta _{\text{d1}}>\eta _{\text{d2}}}\) and underestimated when \({\eta _{\text{d1}}<\eta _{\text{d2}}}\). When \({\eta _{\text{d1}}<\eta _{\text{d2}}}\), the estimated excess noise quickly becomes negative, this happens because, in this situation, the excess noise is dominated by the sixth term in (18), which itself becomes negative when \({\eta _{\text{d1}}<\eta _{\text{d2}}}\). Recall that in our case the only excess noise contributions are due to noise originating in Alice’s transmission system and due to the interference noise between Alice’s signal and Bob’s LO, in the presence of other, likely higher, channel noise contributions, the excess noise would not take a negative value, but would rather have a reduced value when compared to a balanced system. Additionally, for \({\eta _{\text{B}}=0.5}\), when \({\eta _{\text{d1}}<\eta _{\text{d2}}}\) the excess noise estimate will decrease until it reaches a minimum and when \({\eta _{\text{d1}}>\eta _{\text{d2}}}\) it increases until it reaches a maximum. However, this curve is not symmetric, with the minimum value observed not having the same absolute value as the maximum observed value. Meanwhile, when \({\eta _{\text{B}}=0.45}\), the excess noise estimate will exhibit a minimum with a lower value and located at a lower value of \({\eta _{\text{d1}}-\eta _{\text{d2}}}\) and a maximum with a lower value located at a higher value of \({\eta _{\text{d1}}-\eta _{\text{d2}}}\), when compared to the values for \({\eta _{\text{B}}=0.5}\). Conversely, when \({\eta _{\text{B}}=0.55}\), the excess noise estimate will exhibit only the maximum observed when \({\eta _{\text{d1}}>\eta _{\text{d2}}}\), having a higher value and being located at a greater value of \({\eta _{\text{d1}}-\eta _{\text{d2}}}\), again when compared to the values for \({\eta _{\text{B}}=0.5}\). These three curves show a very asymmetric dependency of the excess noise with the photodetector imbalances, this asymmetry is again due to the contribution of the sixth term in (18), which is itself asymmetric, and due to the excess noise’s dependency on the estimated channel transmission, shown in Fig. 3 (a), which will cause the estimated excess noise values when \({\eta _{\text{d1}}-\eta _{\text{d2}}}<0\) to have a higher absolute value than the ones estimated when \({\eta _{\text{d1}}-\eta _{\text{d2}}}>0\).
The secret key rate can again be estimated for the estimated values of transmission and excess noise following (23), yielding the results presented in Fig. 3 (d). We can see from Fig. 3 (d) that, for both \({\eta _{\text{B}}=0.5}\) and \({\eta _{\text{B}}=0.45}\), the estimated secret key rate will, generally, increase as \({\eta _{\text{d1}}-\eta _{\text{d2}}}\) increases, apart from a short decreasing region. Meanwhile, when \({\eta _{\text{B}}=0.55}\), the estimated secret key rate will always increase as \({\eta _{\text{d1}}-\eta _{\text{d2}}}\) increases. The evolution of the estimated key rate in function of the photodiode imbalance is dominated by the estimated value for the channel transmission, shown in Fig. 3 (a), which increases linearly with \({\eta _{\text{d1}}-\eta _{\text{d2}}}\). The regions where the growth of the estimated key rate slows down, and in the case of \({\eta _{\text{B}}=0.5}\) and \({\eta _{\text{B}}=0.45}\) stops, are due to the contribution of the excess noise, whose maximum and minimums roughly coincide with these regions. Depending on the exact position in the x-axis, this means that these combined beam-splitter and photodiode quantum efficiencies will result in either an over or under estimation of the secret key rate. In the scenario of an overestimation of the secret key rate, this will pose a security risk as Alice and Bob will distillate bits for the key at a rate higher than the channel parameters would allow for a secure key, while in the case of an underestimation there will be lost performance, as Alice and Bob will discard more bits than they had to.