In Figure C1 (a), we start the conference key rate optimized for N- 5 parts (equation (C.1), volume lines) based on loss in each quantum channel, for different fixed values of the q parameter and a fixed number of input ports (exit) of the beam divider: M-5. We also expect direct transmission (4.1) for the same number of games. The optimal number of parts in each subset depends on the loss and value of q, and it is shown in Figure C1 (b). 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). We find that the implementation of a truly multi-party scheme could be optimal, especially in the case of minor losses, i.e. if the common state of the parties is well brought together by a multi-party state W. We optimize the key conference rate achieved by N-5 parties on the cardinality of the fractions of the parties that perform the CKA at once.

The different keys defined within each subgroup (which can consist of two, three or five parts) are then used to encode the final conference key. As stated at the end of Section 1, the confused resource used to distill the secret key is a noisy N-class state of N qubits [42], which is selected after single photon interference in the central node. In fact, optimizing the CKA key set (section 4) on the q setting, which laid the initial qubit-photon state overlay, always gives values close to 1. This means that the quantum signal sent by the parties is highly unbalanced in relation to the vacuum. Thus, events where one of the detectors clicks are mainly caused by the arrival and detection of a photon. However, due to the balanced overlay generated by the multiport beam, the recognized photon can be sent by any party with the same probability. Because the photon is first involved in the qubit in the state, the qubit state, which depends on detection, is a consistent overlay of states where one qubit is in condition and where all the others are in condition, i.e. the state mentioned in the W class.