# Missed Detection of Entanglement in Two-Mode Squeezed States Based on the Inseparability Criterion
## 3 Experimental and Results  
Our experimental validation was conducted through a NOPA system. Owing to the symmetry of the Hamiltonian, the state generated by NOPA is a fully symmetric state, implying that the elements of the covariance matrix satisfy conditions, specifically \(n=m\) and \(c_{1}=-c_{2}\). Figure 3 illustrates the experimental setup employed to generate the two-mode squeezed state. The NOPA comprises an \(\alpha\)-cut type-II phase-matched KTiOPO4 (KTP) crystal (cut at \(1^{\circ}\)) and a plano-concave mirror. The front surface of the crystal (S1) is highly reflective (HR) at 1080 nm and has a transmission rate of T = 18% at 540 nm, serving as the input coupling mirror. The rear surface (S2) is anti-reflective (AR) coated for both 1080 nm and 540 nm. The plano-concave mirror has a curvature radius of 50 mm and acts as the output coupling mirror, being highly reflective at 540 nm and having a transmission rate of 12.5% for 1080 nm light. The system utilizes 1080 nm infrared light, which is noise-reduced by a three-mirror ring-cavity infrared mode cleaner (MC1), and 540 nm green light, which is noise-reduced by a green light mode cleaner (MC2), as the input and pump beams, respectively. The NOPA cavity is operated below its threshold, and when the relative phase between the pump and input beams is locked to the parametric deamplification state, the output port emits a two-mode squeezed state with orthogonal  
Figure 2: Region plots illustrating the impact of entanglement asymmetry on entanglement criteria. (**a**) Asymmetry in correlated orthogonal components: with \(n=m=3\), the entanglement criterion is shown as it evolves with respect to \(\xi_{1}\) and \(\xi_{2}\); (**b**) asymmetry in entanglement modes: with \(\xi_{1}=-\xi_{2}=1/\sqrt{2}\), the evolution of the entanglement criterion is depicted with respect to \(n\) and \(m\). The orange area represents the entanglement region defined by the inseparability criterion, while the blue area delineates the entanglement range according to the PPT criterion. The green area indicates un-physical covariance matrices. polarizations (signal field \(a\) and idler field \(b\)). This two-mode squeezed state exhibits anti-correlation in the quadrature amplitudes and positive correlation in the quadrature phases. In the experiment, the two sub-modes \(a\) and \(b\) of the two-mode squeezed state are separated using a polarizing beam splitter (PBS) and then directed to a balanced homodyne detection system for entanglement measurement. To achieve a lower pump threshold, the NOPA cavity is designed to be triply resonant for the pump, signal, and idler modes. This is experimentally realized by fine-tuning the phase-matching temperature and adjusting the transverse position of the crystal. During measurements, the system is operated at approximately 70% of the threshold power to avoid undesirable non-Gaussian effects. The experiment demonstrates a squeezing of \(-4.95\) dB and an anti-squeezing of \(6.53\) dB at an analysis frequency of \(2\) MHz (see Figure 4). The upper trace represents the standard quantum noise limit (QNL), corresponding to approximately \(-53.8\) dBm, which is obtained by blocking the signal light and measuring only the vacuum fluctuations. These measurements were obtained using a balanced homodyne detection system with a total detection efficiency of up to 85%. Complete measurements of the quadrature components allowed for the derivation of the covariance matrix, enabling a comparison between the PPT criterion and the inseparability criterion. The homodyne detection system was calibrated by adjusting the local oscillator (LO) power and balancing the photodetectors to ensure optimal interference and signal-to-noise ratio. Phase stabilization between the signal and LO beams was achieved using a phase-locked loop (PLL). Shot noise levels were calibrated by blocking the signal beam, and electronic noise was confirmed to be at least 10 dB below shot noise.