# Missed Detection of Entanglement in Two-Mode Squeezed States Based on the Inseparability Criterion
## 2 Theoretical Analysis
### Missed Detection of Entanglement  
The covariance matrix, as a vital representation of quantum state statistical information, provides a comprehensive statistical description of the correlation characteristics of two-mode squeezed states. However, it does not directly reveal the separability or entanglement of a state, nor does it quantify the degree of entanglement, necessitating the use of specific entanglement criteria. Inseparability criteria are widely employed in optical experiments, offering a straightforward and efficient tool for investigating the entanglement properties of continuous-variable quantum systems. The criteria typically equate separability with the positivity of the P-function. For states characterized by Gaussian noise distributions and symmetric correlations in conjugate quadrature components, the criteria can be linked  
Figure 1: The diagram illustrates the asymmetry of a two-mode squeezed state, where the asymmetry of the associated orthogonal components and the asymmetry of the entanglement modes are significantly manifested in the preparation and transmission stages, respectively. to measurable correlations [6]. Specifically, for two-mode squeezed states, the inequality takes the form of  
\[Ent_{Inse}=\Delta^{2}X^{+}_{a\pm b}+\Delta^{2}X^{-}_{a\pm b}<1 \tag{4}\]  
where \(\Delta^{2}X^{+}_{a\pm b}\) represents the minimum value of the variance of the sum or difference of operators between beams \(a\) and \(b\), which has been normalized, i.e., \(\Delta^{2}X^{+}_{a\pm b}=\min\Bigl{\langle}\bigl{(}\delta X^{+}_{a}\pm\delta X ^{+}_{b}\bigr{)}^{2}\Bigr{\rangle}/4\). The elements of the covariance matrix can be further utilized to calculate these variances:  
\[Ent_{Inse}=\frac{1}{2}[n+m-(|\epsilon_{1}|-|\epsilon_{2}|)]<1 \tag{5}\]  
The value of \(Ent_{Inse}\) directly reflects the strength of entanglement in the quantum state. The closer the value is to zero, the higher the degree of entanglement. It is important to note that the inseparability criterion is merely a sufficient condition for entanglement. Specifically, satisfying this inequality confirms the existence of entanglement. However, this criterion is not necessary--satisfying inequality (4) does not guarantee that the state is separable. The inseparability criterion fails to capture all entangled states, meaning certain entangled states may not satisfy this criterion, posing a risk of missed detections. This could result in misjudgments or errors in practical quantum information tasks, thereby adversely affecting the efficiency and accuracy of these tasks. We aim to explore the potential risk of missing entangled states when applying the inseparability criterion to asymmetric two-mode squeezed states. To achieve this, we will compare the inseparability criterion with the PPT criterion, which is more reliable in detecting entanglement. The PPT criterion serves as a sufficient and necessary condition for two-mode squeezed states, determining the separability of a state by the positive definiteness of the partial transpose matrix of the two-mode Gaussian state. Specifically, the PPT criterion quantifies the entanglement properties by calculating the minimum symplectic eigenvalue of the covariance matrix [15],  
\[Ent_{PPT}=\sqrt{\frac{\Gamma-\sqrt{\Gamma^{2}-4\text{det}\sigma}}{2}}<1 \tag{6}\]  
where \(\Gamma=\text{det}\mathbf{A}+\text{det}\mathbf{B}-2\text{det}\mathbf{C}\). If the inequality holds, the two-mode Gaussian state is considered entangled; otherwise, it is separable. In Figure 2, we visually compare the inseparability and PPT criteria in defining the entanglement region under two scenarios: asymmetry in the intermodal correlation of orthogonal components and asymmetry in the entanglement modes.