This paper presents necessary and sufficient conditions for the separability of mixed quantum states. The authors show that for 2×2 and 2×3 systems, the positivity of the partial transposition of a state is both necessary and sufficient for its separability. However, this is not generally true for higher-dimensional systems. The paper also discusses the implications of these findings for quantum information theory, including quantum computation and teleportation. The authors emphasize that mixed states, which are more common in experimental settings, can exhibit quantum behavior even though they are not pure states. They provide examples to illustrate the utility of the separability criterion and discuss its relationship with entropy inequalities. The paper concludes that while the partial transposition criterion is effective for 2×2 and 2×3 systems, it is not sufficient in general. The authors also highlight the importance of distinguishing between classical and quantum states in the context of mixed states. The paper is a significant contribution to the understanding of quantum separability and its practical applications.This paper presents necessary and sufficient conditions for the separability of mixed quantum states. The authors show that for 2×2 and 2×3 systems, the positivity of the partial transposition of a state is both necessary and sufficient for its separability. However, this is not generally true for higher-dimensional systems. The paper also discusses the implications of these findings for quantum information theory, including quantum computation and teleportation. The authors emphasize that mixed states, which are more common in experimental settings, can exhibit quantum behavior even though they are not pure states. They provide examples to illustrate the utility of the separability criterion and discuss its relationship with entropy inequalities. The paper concludes that while the partial transposition criterion is effective for 2×2 and 2×3 systems, it is not sufficient in general. The authors also highlight the importance of distinguishing between classical and quantum states in the context of mixed states. The paper is a significant contribution to the understanding of quantum separability and its practical applications.