# Missed Detection of Entanglement in Two-Mode Squeezed States Based on the Inseparability Criterion
## 2 Theoretical Analysis  
### Asymmetry of Two-Mode Gaussian States  
The two-mode squeezed state is a fundamental Gaussian quantum state in quantum optics, typically generated by a type-II nondegenerate optical parametric amplifier (NOPA) or by combining two independent single-mode squeezed fields using a 50/50 beam splitter. Mathematically, this state can be described by the symplectic eigenform of its covariance matrix, denoted as \(\sigma_{ij}=\big{\langle}\big{(}X_{i}X_{j}+X_{j}X_{i}\big{)}\big{\rangle}/2- \big{\langle}X_{i}\big{\rangle}-\big{\langle}X_{j}\big{\rangle}\), where \(X\equiv\big{(}X_{a}^{+},X_{a}^{-},X_{b}^{+},X_{b}^{-}\big{)}\) represents the quadrature components of the field. Here, the subscripts \(a\) and \(b\) denote the two modes, as illustrated in Figure 1, and the superscripts \(+\) and \(-\) denote the quadrature amplitude and quadrature phase components, respectively. These quadrature components are related to the annihilation operator \(a_{i}\) and creation operator \(a_{i}^{+}\) of the field as \(X_{i}^{+}=a_{i}+a_{i}^{+}\) and \(X_{i}^{-}=i\big{(}a_{i}^{+}-a_{i}\big{)}\),  
\[\sigma=\begin{bmatrix}\mathbf{A}&\mathbf{C}\\ \mathbf{C}^{T}&\mathbf{B}\end{bmatrix}=\begin{bmatrix}n&0&c_{1}&0\\ 0&n&0&c_{2}\\ c_{1}&0&m&0\\ 0&c_{2}&0&m\end{bmatrix} \tag{1}\]  
where the submatrices \(\mathbf{A}\) and \(\mathbf{B}\) correspond to the autocovariance matrices of the two subsystems, while the off-diagonal elements \(\mathbf{C}\) represent the cross-covariance matrix between the two subsystems [39]. The relevant quantities \(n\), \(m\), \(c_{1}\), and \(c_{2}\) are determined by four local symplectic invariants: \(\text{det}\sigma=\big{(}nm-c_{1}^{2}\big{)}\big{(}nm-c_{2}^{2}\big{)}\), \(\text{det}\mathbf{A}=n^{2}\), \(\text{det}\mathbf{B}=m^{2}\), and \(\text{det}\mathbf{C}=c_{1}c_{2}\). Notably, the parameters are not arbitrary but must be subject to specific physical constraints [22]:  
\[\sigma+\frac{i}{2}\mathbf{\omega}\oplus\mathbf{\omega}\geq 0 \tag{2}\]where \(\mathbf{\omega}\equiv\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\). These constraints can be further elucidated as a function of the symplectic eigenvalues of the matrix elements, providing a deeper understanding of the state's properties,  
\[\text{det}\mathbf{A}+\text{det}\mathbf{B}+2\text{det}\mathbf{C}\leq 4\text{det}\mathbf{\sigma}+1 \tag{3}\]  
A state is termed symmetric when the covariance matrix satisfies \(n=m\). For an ideal symmetric state, the parameters in the covariance matrix satisfy both \(n=m\) and \(c_{1}=-c_{2}\).  
However, in practical applications, imperfections such as equipment noise and operational inefficiencies often result in asymmetry in two-mode squeezed states [40]. As illustrated in Figure 1, this asymmetry manifests primarily in two aspects: asymmetry in the orthogonal components of inter-mode correlations and asymmetry between the entangled modes. The former refers to the difference between the inter-mode covariance components \(c_{1}\) and \(c_{2}\). This asymmetry typically arises from inconsistencies in the degree of squeezing, squeezing direction, or phase differences between the two squeezed beams (SQ1 and SQ2) used to generate the two-mode squeezed state during the state preparation process [23]. Mode asymmetry, on the other hand, refers to the difference between the local covariance matrices \(n\) and \(m\) of the two modes. This asymmetry is often a result of the state transmission and detection processes, commonly encountered in quantum information protocols involving asymmetric noise and loss channels, such as quantum illumination and one-sided device-independent quantum key distribution. Asymmetry can significantly impact the results of entanglement criteria, making it crucial to understand and analyze the effects in practical systems.