# Missed Detection of Entanglement in Two-Mode Squeezed States Based on the Inseparability Criterion
## 4 Conclusions  
This paper delves into the utilization of the inseparability criterion in assessing two-mode squeezed states, with a particular focus on analyzing the phenomenon of entanglement detection failures caused by entanglement asymmetry. To this end, we conducted a comparative analysis with the more stringent necessary and sufficient condition--the  
Figure 5: Experimental results of entanglement decoherence in lossy channel. (**a**) Symmetric lossy channel; (**b**) asymmetric lossy channel. Figure 6: Effect of channel attenuation on the inseparability criterion for mixed entanglement state: (**a**) symmetric lossy channel; (**b**) asymmetric lossy channel. PPT criterion, both theoretically and experimentally, investigating the instances where the inseparability criterion fails to detect entanglement. Experimentally, we employed a sub-threshold NOPA to generate two-mode squeezed states and introduced optical losses to induce asymmetry in the entangled modes, thereby validating the limitations of the inseparability criterion. Our results reveal that in the presence of symmetric entanglement, the inseparability criterion aligns with the PPT criterion, constituting a necessary and sufficient condition for two-mode Gaussian states. However, under asymmetric entanglement conditions, significant disparities emerge between the two criteria, with the inseparability criterion only serving as a sufficient condition. Consequently, exclusive reliance on the inseparability criterion may result in missed entanglement detection and erroneous entanglement assessments, with the misjudgment rate escalating alongside the intensification of entanglement asymmetry. Hence, when confronted with highly asymmetric two-mode squeezed states, a more prudent application of the inseparability criterion is warranted. This may involve adjusting the covariance matrix weights, incorporating compensation factors into the inseparability criterion calculation, or adopting stricter criteria to ensure accurate entanglement state determination. Additionally, exploring novel neural network-based methods for continuous-variable entanglement detection [41] offers promising avenues. Recent advancements in discrete-variable entanglement detection, such as machine learning approaches [42] and complex-valued neural networks [43], provide valuable insights and techniques that could potentially be adapted to continuous-variable systems, further advancing the field. Future research could focus on refining entanglement detection methodologies to address the challenges posed by asymmetric entanglement. This includes developing hybrid criteria that combine traditional mathematical approaches with emerging machine learning-based techniques, which hold significant promise for handling complex quantum systems. Practical applications, such as quantum communication and quantum computing, where secure entanglement distribution and entanglement resource management are critical, stand to benefit greatly from these advancements. For instance, in quantum communication, our findings highlight the importance of robust entanglement detection in ensuring the security and reliability of quantum key distribution protocols, particularly in real-world scenarios involving asymmetric entanglement. In quantum computing, accurate identification of entanglement resources could play a pivotal role in optimizing quantum gate operations and error correction schemes, thereby enhancing computational efficiency. Investigating the role of asymmetric entanglement in quantum networks and sensors also represents an exciting avenue for exploration. In conclusion, this study offers vital insights into the application and underlying principles of the inseparability criterion in quantum information tasks, while also highlighting its potential limitations. Future quantum information science research should prioritize the impact of asymmetric entanglement, continually refining entanglement detection and measurement methodologies.