## 4 Simulation Results  
In this section, we consider the CV quantum conferencing with entanglement in the middle against collective attacks and give the simulation results of our security analysis. All of the quantum channelshave the same transmissivity \(\eta\), which is mapped into a fiber distance \(l\) in km, using \(\eta:=10^{l/50}\). Different conditions of thermal noise and number of users are taken into account.  
In Figure 2, we plot the secret key rate of our quantum conferencing scheme versus the maximum distance of the quantum link in our network. The solid blue lines, small dashed red lines and large dashed black lines refer to \(N=3\), \(N=10\) and \(N=100\) network users of our quantum conferencing scheme, respectively. As shown in Figure 2, the scheme can reach a network radius of 2 km when the number of users is less than 10. When the number of users increases to 100, the scheme can still provide a quantum conferencing network with a longest transmission distance of over 500 m (520 m when thermal noise is set to 0.05, and 575 m when thermal noise is set to 0). Furthermore, the curves show that, with a network radius of 480 m, 100 users can establish secure conferencing communication at about 0.1 bit per pulse. The conferencing key rate can reach 2.5 Mbits per second based on a CV QKD network with a clock of 25 MHz [27].  
Figure 3 presents the relationship between number of users and the maximum network distance when the secret conferencing key rate is asymptotic to 0, and the thermal noise is set to \(\bar{n}=0.05\) in (a) and \(\bar{n}=0\) in (b), respectively. As can be seen from Figure 3, there is a trade-off between the number of network users and the maximum transmission distance. Moreover, the conferencing scheme can achieve secure communication for 300 users on a building-size scale (about a radius of 150 m), or for dozens of users between multiple buildings (within a radius of 500 m or more). This result shows that the quantum conferencing with entanglement in the middle scheme can be applied to more scenarios than the MDI-based high-rate quantum conferencing scheme [19], which provides secure conference for 50 users within a radius of 40 m.  
We consider different measurement methods that the network perform on their received modes. In the key rate equation Equation (46), we replace the mutual information \(I(B_{i}:B_{j})\) that was derived in Equation (28) with the result in Equation (32), and give the simulation results in Figure 4. Obviously, the scheme with homodyne detection has better performance of the secure conferencing key rate than the scheme with heterodyne detection. As we mentioned in Section 2, the \(p\)-quadratures of the entangled GHZ state have correlations of \(\hat{\mathbb{a}}_{k}^{rp}-\hat{\mathbb{a}}_{k^{\prime}}^{rp}\to 0\) for any \(k,k^{\prime}=1,\cdots,N\), which is utilized for the conferencing key generation of our scheme. As a result, we select homodyne detection as the measurement method for the network users.  
Figure 2: The performance analysis of the entanglement-based multipartite CV quantum conferencing scheme in terms of the derived secret key rate. The curves represent the conferencing key rate as a function of the radius of the network, in conditions of \(N=3\) (solid blue line), \(N=10\) (small dashed red line) and \(N=100\) (large dashed black line) of users, respectively. The thermal noise is set to \(\bar{n}=0.05\) in (a) and \(\bar{n}=0\) in (b), respectively.