## 2 A Quantum Conferencing Scheme with Entanglement in the Middle  
The proposed entanglement-based CV quantum conferencing network is consists of an arbitrary number (\(N\)) of legitimate users, an entanglement source in the middle and links from the source to the users, which include classical networks and quantum channels. The entanglement source prepares an \(x\)-quadrature-squeezed state \(\hat{\mathtt{a}}_{1}\) and \(N-1\)\(p\)-quadrature-squeezed state \(\hat{\mathtt{a}}_{2,\cdots,N}\), which are illustrated in Figure 1. As shown in Figure 1, the source combines the above \(N\) states by a sequence of beam splitters with decreasing transmissivities \(T_{k}=1-1/(N-k+1)\) for \(k=1,\cdots,\ N-1\). The output modes \(\hat{\mathtt{a}}^{\prime}_{1,\cdots,N}\) after the combinations by the beam splitters are entangled in a multipartite GHZ state which satisfies the relations of \(x\)-quadratures \(\sum_{k=1}^{N}\hat{\mathtt{a}}^{\prime x}_{k}\to 0\) and \(p\)-quadratures \(\hat{\mathtt{a}}^{\prime p}_{k}-\hat{\mathtt{a}}^{\prime p}_{k^{\prime}}\to 0\) for any \(k,k^{\prime}=1,\cdots,\ N\). This multipartite GHZ state is utilized for generating secret key in our quantum conferencing scheme.  
The source sends the \(N\) modes of the GHZ state to \(N\) legitimate parties named as Bob\({}_{1,\cdots,N}\), respectively. The entangled modes travel through \(N\) noisy quantum channels toward the users, respectively. For simplicity, we consider a symmetric configuration of the quantum conferencing scheme that all quantum channels from the source to the users have the same length. In a practical scenario, the transmissivity and thermal noise of the longest quantum channel are selected as the reference to the network configuration. The legitimate parties receive the \(N\) entangled modes \(\hat{\mathtt{a}}^{\prime\prime}_{1,\cdots,N}\) transmitted through the quantum channels and perform homodyne detection on them, respectively. With the measurement results, the users share mutual information which can generate a secret key for network communication after post-processing procedures. For a quantum conferencing scheme, the result of the \(i\)th user is used for reference to the reconciliation procedures, which let other users generate secret keys that can establish secure communication with the \(i\)th user, respectively. The CV quantum conferencing protocol with an entanglement source in the middle goes as follows.  
1. The source in the middle prepares a CV GHZ state \(\rho_{1}\) of \(N\) entangled modes \(\hat{\mathfrak{a}}^{\prime}_{1,\cdots,N}\), and sends them to \(N\) legitimate parties \(\text{Bob}_{1,\cdots,N}\) through quantum channels with the same transmissivity \(\eta\), respectively.
2. After travelling through the noisy channels, \(N\) entangled modes \(\hat{\mathfrak{a}}^{\prime\prime}_{1,\cdots,N}\) of the multipartite GHZ state \(\rho_{2}\) are received by \(\text{Bob}_{1,\cdots,N}\), respectively.
3. The users perform homodyne detections on the \(p\)-quadratures of the received modes, respectively. Then, they perform quadrature measurement.
4. With the results of the measurement, the users use a public channel to complete following procedures as parameter estimation, information reconciliation and privacy amplification.