Figure 4

Average number of signals, \(\left\langle N\right\rangle {| }_{L=0}\), that Alice needs to send Bob to collect a data block size equal to \({n}_{{\rm{S}}{\rm{H}}}\) when using ideal photon sources, as a function of the detection and coupling efficiency \({\eta }_{{\rm{c}},{\rm{d}}}\) at \(L=0\) km. As in Eqs. 5 and 6), in this figure we disregard dark counts because their effect at \(L=0\) km is negligible. Also, we set the free experimental and security parameters to those values that optimise the secret key rate given by Fig. 2(b). The figure considers three different data block sizes, i.e., \({n}_{{\rm{S}}{\rm{H}}}=1{0}^{7}\), \({n}_{{\rm{S}}{\rm{H}}}=1{0}^{9}\) and \({n}_{{\rm{S}}{\rm{H}}}=1{0}^{11}\). All the plots are cut at the value of \({\eta }_{{\rm{c}},{\rm{d}}}\) for which the resulting secret key rate is below the threshold value of \(1{0}^{-10}\). We note that, since in the case of the ESR the value of \(\left\langle N\right\rangle {| }_{L=0}\) does not depend on any parameter to be optimised, the cases \({S}_{1}\) and \({S}_{2}\) only differ in the minimum \({\eta }_{{\rm{c}},{\rm{d}}}\) that still provides \(K\ge 1{0}^{-10}\), which can be extracted from Fig. 2. In the case of the PQA, \(\left\langle N\right\rangle {| }_{L=0}\) depends on the transmittance \(t\) to be optimised, and therefore the cases \({S}_{1}\) and \({S}_{2}\) differ more from each other.