## Security Analysis  
The entangled source generates the initial state \(\rho_{0}\) of an \(x\)-quadrature squeezed mode \(\hat{\mathtt{a}}_{1}\) and \(N-1\)\(p\)-quadrature squeezed modes \(\hat{\mathtt{a}}_{2,\cdots,N}\). The covariance matrices of the initial modes are defined as  
\[\mathbf{a}_{1}= \begin{pmatrix}e^{-2r}&0\\ 0&e^{2r}\end{pmatrix} , \tag{1}\]  
and  
\[\mathbf{a}_{2,\cdots,N}= \begin{pmatrix}e^{2r}&0\\ 0&e^{-2r}\end{pmatrix} . \tag{2}\]  
The quadrature vector \(\xi_{0}\) of \(\rho_{0}\) is denoted as  
\[\xi_{0}= \begin{pmatrix}\hat{\mathtt{a}}_{1},\hat{\mathtt{a}}_{2},\cdots,\hat{ \mathtt{a}}_{N}\end{pmatrix} ^{T}. \tag{3}\]  
To represent the \(2N\) quadratures of the \(N\) modes, \(\xi_{0}\) is restructured into the following form,  
\[\xi_{1}= \begin{pmatrix}\hat{\mathtt{a}}_{1}^{x},\hat{\mathtt{a}}_{2}^{x}, \cdots,\hat{\mathtt{a}}_{N}^{x},\hat{\mathtt{a}}_{1}^{p},\hat{\mathtt{a}}_{2}^ {p},\cdots,\hat{\mathtt{a}}_{N}^{p},\end{pmatrix} ^{T}. \tag{4}\]  
After generating the initial modes, the source firstly combines \(\hat{\mathtt{a}}_{1}\) and \(\hat{\mathtt{a}}_{2}\) by a beam splitter of transmissivity \(T_{1}\). In the next \(N-2\) steps, one of the output mode from the \(k\)th combination and an initial mode \(\hat{\mathtt{a}}_{k+1}\) are combined by a beam splitter of transmissivity \(T_{k}\), for \(k=2,\cdots,N-1\). This series of combinations produces \(N\) output modes \(\hat{\mathtt{a}}_{1,\cdots,N}^{\prime}\), which are entangled in a GHZ state. The transmissivities of the \(N-1\) beam splitters of the entangled source is denoted by \(T_{k}=\frac{N-k}{N-k+1}\), for \(k=1,2,\cdots,N-1\). The output modes is described by the linear transformations on \(x\)-quadratures  
\[\hat{\mathtt{a}}_{1}^{\prime x} \rightarrow \frac{\hat{\mathtt{a}}_{1}^{x}}{\sqrt{N}}+\sum_{j=2}^{N}\frac{ \hat{\mathtt{a}}_{j}^{x}}{(N+2-j)(N+1-j)}, \tag{5}\] \[\hat{\mathtt{a}}_{k}^{\prime x} \rightarrow \frac{\hat{\mathtt{a}}_{1}^{x}}{\sqrt{N}}+\sum_{j=2}^{k}\frac{ \hat{\mathtt{a}}_{j}^{x}}{(N+2-j)(N+1-j)}\quad\text{for $k=2,\cdots,N$}, \tag{6}\]  
and the analogous linear transformations on \(p\)-quadratures. Moreover, the covariance matrices of the beam splitters are denoted by  
\[\mathbf{T}_{k}= \begin{pmatrix}\sqrt{\frac{N-k}{N+1-k}}&-\frac{1}{\sqrt{N+1-k }}\\ \frac{1}{\sqrt{N+1-k}}&\sqrt{\frac{N-k}{N+1-k}}\end{pmatrix} \quad\text{for $k=1,\cdots,N-1$}. \tag{7}\]  
We use symplectic matrix \(\mathbf{R}\) to describe the combinations of the beam splitters \(T_{1,\cdots,N-1}\). The elements of \(\mathbf{R}\) are given by \[\mathbf{R}_{11} =\frac{1}{\sqrt{N}}, \tag{8}\] \[\mathbf{R}_{1j} =\frac{1}{\sqrt{(N+2-j)(N+1-j)}}\quad\text{for }j=2,\cdots,N,\] (9) \[\mathbf{R}_{jk} =-\mathbf{R}_{1k}\quad\text{for }j=2,\cdots,N\text{ and }k=1, \cdots,j-1,\] (10) \[\mathbf{R}_{kk} =\sqrt{\frac{N+1-k}{N+2-k}}\quad\text{for }k=2,\cdots,N. \tag{11}\]  
The covariance matrix (CM) of the initial state \(\rho_{0}\) is denoted by  
\[\mathbf{V}_{0}= \begin{pmatrix}\mathbf{V}_{x}&\mathbf{0}\\ \mathbf{0}&\mathbf{V}_{p}\end{pmatrix} , \tag{12}\]  
in which \(\mathbf{V}_{x}\) and \(\mathbf{V}_{p}\) are the CM of the \(x\)-quadratures and the \(p\)-quadratures of \(\rho_{0}\) in terms of the quadrature vector \(\xi_{1}\), respectively. The CM \(\mathbf{V}_{x}\) and \(\mathbf{V}_{p}\) are denoted by the following forms  
\[\mathbf{V}_{x}= \begin{pmatrix}e^{-2r}&\mathbf{0}\\ \mathbf{0}&e^{2r}\mathbf{I}_{N-1}\end{pmatrix} ,\quad\mathbf{V}_{p}= \begin{pmatrix} e^{2r}&\mathbf{0}\\ \mathbf{0}&e^{-2r}\mathbf{I}_{N-1}\end{pmatrix} , \tag{13}\]  
where \(r\) refers to the squeeze factor of the initial squeezed states. The source combines the modes of \(\rho_{0}\) by the beam splitters, producing the entangled modes of a GHZ state \(\rho_{1}\), which is described by the following CM  
\[\mathbf{V}_{1}= \begin{pmatrix}\mathbf{V}_{x}^{\prime}&\mathbf{0}\\ \mathbf{0}&\mathbf{V}_{p}^{\prime}\end{pmatrix} =\mathbf{R}\cdot\mathbf{V}_{0} \cdot\mathbf{R}^{T}, \tag{14}\]  
where  
\[ \begin{pmatrix}\mathbf{V}_{x}^{\prime}\end{pmatrix} _{ij}=