# Missed Detection of Entanglement in Two-Mode Squeezed States Based on the Inseparability Criterion
These criteria determine whether a system is in an entangled or separable state by measuring the expectation values of specific observables. Among these, the inseparability criterion, renowned for its simplicity and experimental feasibility, has been extensively employed in quantum optics experiments. Introduced by [PERSON] et al. [17] and [PERSON] [15], the criterion relies on measuring the total variance of EPR-type operator pairs and comparing it with the standard quantum limit to ascertain entanglement. Researchers have leveraged this criterion to explore the dynamical behavior of entangled states in complex environments, such as noisy and lossy channels [18; 19; 20; 21]. Notably, studies like those by [PERSON] et al. have validated the robustness of entangled states under strong decoherence using various quantum indicators, including teleportation fidelity, quantum discord, mutual information, and the inseparability criterion [22]. [PERSON] et al. investigated the effects of symmetric decoherence on the EPR paradox and inseparability criterion, revealing their qualitative disparities [23]. Furthermore, the inseparability criterion has been extended to assess intricate multi-mode Gaussian quantum entangled states, such as four-mode cluster states [24] and hyperentangled states simultaneously possessing orbital angular momentum and spin angular momentum [25], highlighting its versatility. While previous research primarily focused on symmetric channel decoherence, the increasing importance of asymmetric decoherence phenomena in quantum communication and computation underscores the need for closer examination under asymmetric conditions. [PERSON] has addressed the challenges posed by asymmetric entanglement and channel losses in quantum illumination systems, utilizing the inseparability criterion to analyze the evolution of non-classical correlations [26; 27; 28; 29; 30]. [PERSON] et al. [31] and [PERSON] et al. [32] have also contributed to this field by characterizing the transmission evolution of entangled fields in optical fibers and analyzing the impact of state asymmetry on entanglement properties and robustness [31; 32]. Research by [PERSON] et al. further reinforces this trend, emphasizing the significance of exploring quantum entanglement under asymmetric decoherence conditions [33]. The inseparability criterion stands as a sufficient condition for assessing entanglement, proficiently identifying its presence in diverse scenarios. Nevertheless, its non-necessity poses a limitation: in situations characterized by significant asymmetry within the system, particularly those involving asymmetric decoherence or intrinsic quantum state asymmetry, it may fall short of fully and precisely quantifying the extent of entanglement, thereby posing a risk of missed entanglement detections and ultimately leading to an underestimation of the entanglement degree. This limitation transcends specific applications like quantum illumination for target recognition and ranging [34; 35], one-sided device-independent quantum key distribution [36], and correlated quadrature asymmetries in entanglement [37; 38], posing a tangible threat to the operational integrity of quantum technologies. The complexities inherent in asymmetric quantum systems present a significant challenge to the inseparability criterion, impairing its capacity to fully capture the intricacies of entangled states. As a result, there exists a risk of underestimating the degree of entanglement and missing crucial detections, which could, in turn, jeopardize the practical implementation of quantum technologies. In quantum communication, missed detections may compromise the security of information transmission, while in quantum computation, they may threaten the accuracy of algorithm execution and the reliability of outcomes. Therefore, it is imperative to rigorously evaluate the strengths and limitations of the inseparability criterion in these contexts.