The paper provides necessary and sufficient conditions for the separability of mixed states in quantum systems. Specifically, it offers a simple criterion for separability in $2 \times 2$ and $2 \times 3$ systems, where the positivity of the partial transposition of a state is both necessary and sufficient. However, this criterion is not generally applicable. The authors demonstrate that for these systems, the positivity of the partial transposition is a sufficient condition, but it is not necessary in higher dimensions. They also provide examples to illustrate the utility of their criterion and discuss the implications for practical applications of quantum inseparability, such as quantum computation and teleportation. The results are based on information-theoretic approaches and the use of positive maps, which are essential for characterizing separable states.The paper provides necessary and sufficient conditions for the separability of mixed states in quantum systems. Specifically, it offers a simple criterion for separability in $2 \times 2$ and $2 \times 3$ systems, where the positivity of the partial transposition of a state is both necessary and sufficient. However, this criterion is not generally applicable. The authors demonstrate that for these systems, the positivity of the partial transposition is a sufficient condition, but it is not necessary in higher dimensions. They also provide examples to illustrate the utility of their criterion and discuss the implications for practical applications of quantum inseparability, such as quantum computation and teleportation. The results are based on information-theoretic approaches and the use of positive maps, which are essential for characterizing separable states.