We observe that the key CKA game can exceed the direct transfer related to sufficiently large losses. This is expected to be the case, as the key rate of the CKA is essentially linearly modulated in relation to the t-transmission of the quantum channel that connects part to the central node, while the direct transmission rate is modulated linearly by the transmission (t2) of the entire channel that connects two parts [33]. However, we note that the CKA`s performance advantage with respect to the direct transmission limit is decreasing for an increasing number of parties. This is because an increase in the number of parts is more damaging to the CKA rate, as it strongly affects the QBER5 than it does to the person directly transferred, where it simply increases the total number of towers that divide the length of the keys. In addition, the presence of dark counts in the detectors prevents the CKA from exceeding the direct transmission limit when the number of parts is too high (see No. 9 cases in Figure 2). In fact, they cause the key rate to drop suddenly in the event of large losses, that is, when the probability of discovering a photon becomes comparable to the probability of an obscure count. In addition, their effect increases with the number of games, because optimizing the key phrase makes it less likely to have a single click in one of the detectors when more parties are involved. Note, however, that if the CKA level takes into account the imperfections of the devices (for example. B dark counts), the direct transmission limit is reached only in the ideal scenario without imperfections. Figure C1.

(a) The optimized conference key rate (equation (C.1) based on loss in the channel connecting part to the central node for different fixed values of q -0.995, 0.998 and 0.999 (up and down). We also draw direct transmission (equation (4.1), dotted line) for five parts. We observe that the optimized key set exceeds the standard CKA, especially in case of significant losses (compared to Case N – 5 in Figure 2), as a smaller number of parties participating in the CKA reduces the negative effect of darkness at once. b) The optimal number of parts in the subsets in which the total number of users (N-5) was divided, based on loss in a quantum channel, for different fixed values of q-0.995, 0.998 and 0.999 (bottom to top).