# Missed Detection of Entanglement in Two-Mode Squeezed States Based on the Inseparability Criterion
## 3 Experimental and Results
The total uncertainty, including detector efficiency and phase fluctuation effects, was estimated to be within \(\pm 0.2\) dB. Figure 4: Measured entanglement spectrum at an analysis frequency of \(2\) MHz with the inseparability criterion. Figure 3: Schematic of the experiment. NOPA: non-degenerate optical parametric amplifier; MC1: mode cleaner; MC2: mode cleaner; HWP: half-wave plate; PBS: polarizing beamsplitter; DBS: dichroic beamsplitter. To introduce asymmetry in the entanglement, we strategically positioned independent beam splitters on the two entangled beams, prior to their arrival at the balanced homodyne detectors. This setup simulates propagation losses inherent quantum channels. These beam splitters were constructed using a half-wave plate and a polarizing beam splitter, enabling us to simulate channel attenuation ranging from 0% to 100% by adjusting the angle of the half-wave plate. The fields \(a^{\prime}\) and \(b^{\prime}\) preceding the balanced homodyne detectors can be mathematically expressed as \(a^{\prime}=\sqrt{\eta_{a}}a+\sqrt{1-\eta_{a}}a_{\text{\tiny{D}}}\) and \(b^{\prime}=\sqrt{\eta_{b}}b+\sqrt{1-\eta_{b}}b_{\text{\tiny{D}}}\), where \(\sqrt{\eta_{a}}\) and \(\sqrt{\eta_{b}}\) represent the channel transmission, and \(a_{\text{\tiny{D}}}\) and \(b_{\text{\tiny{D}}}\) are the vacuum fields introduced into the system due to the lossy channels. The two-mode Gaussian state \(\rho_{a^{\prime}b^{\prime}}\), formed by the fields \(a^{\prime}\) and \(b^{\prime}\), can be effectively characterized by a covariance matrix:  
\[\mathbf{\sigma}^{\prime}= \begin{bmatrix}\eta_{a}n+1-\eta_{A}&0&\sqrt{\eta_{a}\eta_ {b}}c&0\\ 0&\eta_{a}n+1-\eta_{a}&0&-\sqrt{\eta_{a}\eta_{b}}c\\ \sqrt{\eta_{a}\eta_{b}}c&0&\eta_{b}n+1-\eta_{b}&0\\ 0&-\sqrt{\eta_{a}\eta_{b}}c&0&\eta_{b}n+1-\eta_{b}\end{bmatrix}  \tag{7}\]  
For each attenuation, the inseparability value and PPT value are obtained by repeating the complete measurement process of the quadrature components. These entanglement criteria can be further formulated as functions of the channel attenuation, with the inseparability value,  
\[Ent^{\prime}_{Inse} =\Delta^{2}X^{+}_{a^{\prime}\pm b^{\prime}}+\Delta^{2}X^{-}_{a^{ \prime}\pm b^{\prime}} \tag{8}\] \[=\tfrac{1}{2}\big{(}\eta_{a}n+\eta_{b}n+(1-\eta_{a})+(1-\eta_{b}) +2\sqrt{\eta_{a}\eta_{b}}c\big{)}\]  
and PPT value,  
\[Ent^{\prime}_{PPT}=\sqrt{\frac{\Gamma^{\prime}-\sqrt{\Gamma^{\prime 2}-4\text{ det}\mathbf{\sigma}^{\prime}}}{2}} \tag{9}\] \[\Gamma^{\prime}=\left(\eta_{a}n+1-\eta_{a}\right)^{2}+\left(\eta_ {b}n+1-\eta_{b}\right)^{2}+2\eta_{a}\eta_{b}c^{2},\] \[\text{ det}\mathbf{\sigma}^{\prime}=\left[\left(\eta_{a}n+1-\eta_{a} \right)\left(\eta_{b}n+1-\eta_{b}\right)-\eta_{a}\eta_{b}c\right]^{2},\]  
Figure 5 presents the experimental outcomes for entanglement detection utilizing the inseparability criterion. Panels (a) and (b) illustrate the scenarios of symmetric (\(\eta_{a}=\eta_{b}=\eta\)) and asymmetric channel attenuation (\(\eta_{a}=1\) and \(\eta_{b}=\eta\)), respectively. Value 1 represents QNL, acquired by blocking the entangled light and allowing only the local light to enter the balanced homodyne detection system. The red dots and blue squares correspond to experimental measurements based on the inseparability and PPT criterion, respectively, with their respective theoretical fits depicted by matching colored curves. Data was collected at 10 distinct channel attenuation points, spaced at 0.1 intervals. At zero channel attenuation, the initial entanglement degree was \(-4.95\) dB, accompanied by an anti-squeezing of 6.34 dB, indicative of a state purity of 0.73. In the case of symmetric losses, the results from both criteria coincide, revealing a linear decrease in entanglement with the increasing losses, remaining below 1 within a certain range of channel attenuation \([0,1)\). Conversely, under asymmetric losses, as the asymmetry intensifies, the inseparability criterion diverges from the PPT criterion.