Asher Peres proposed a separability criterion for density matrices, showing that a necessary condition for separability is that a matrix obtained by partial transposition of the density matrix has only non-negative eigenvalues. This criterion is stronger than Bell's inequality. A quantum system is separable if its density matrix can be written as a sum of direct products of density matrices for the two subsystems. The separability condition is derived by analyzing the matrix elements of the density matrix and defining a new matrix σ, which is Hermitian and invariant under separate unitary transformations. When the separability condition holds, σ has no negative eigenvalues. This condition is necessary for the existence of the decomposition. For example, in the case of a Werner state, the separability criterion is fulfilled when the singlet fraction x is less than 1/3. This result suggests that the necessary condition might also be sufficient for any ρ. However, for higher dimensions, the condition is not sufficient. Another example is a mixed state introduced by Gisin, where the σ matrix has a negative eigenvalue when x exceeds a certain threshold. This indicates that the separability criterion is more restrictive than Bell's inequality. The weakness of Bell's inequality is due to its limited use of the density matrix. The separability criterion provides a stronger test for inseparability. The criterion is based on the partial transposition of the density matrix and has been shown to be necessary for separability. However, it is not always sufficient. The criterion has been proven for composite systems of dimensions 2×2 and 2×3, but not for higher dimensions. The criterion is more effective than Bell's inequality in detecting inseparability.Asher Peres proposed a separability criterion for density matrices, showing that a necessary condition for separability is that a matrix obtained by partial transposition of the density matrix has only non-negative eigenvalues. This criterion is stronger than Bell's inequality. A quantum system is separable if its density matrix can be written as a sum of direct products of density matrices for the two subsystems. The separability condition is derived by analyzing the matrix elements of the density matrix and defining a new matrix σ, which is Hermitian and invariant under separate unitary transformations. When the separability condition holds, σ has no negative eigenvalues. This condition is necessary for the existence of the decomposition. For example, in the case of a Werner state, the separability criterion is fulfilled when the singlet fraction x is less than 1/3. This result suggests that the necessary condition might also be sufficient for any ρ. However, for higher dimensions, the condition is not sufficient. Another example is a mixed state introduced by Gisin, where the σ matrix has a negative eigenvalue when x exceeds a certain threshold. This indicates that the separability criterion is more restrictive than Bell's inequality. The weakness of Bell's inequality is due to its limited use of the density matrix. The separability criterion provides a stronger test for inseparability. The criterion is based on the partial transposition of the density matrix and has been shown to be necessary for separability. However, it is not always sufficient. The criterion has been proven for composite systems of dimensions 2×2 and 2×3, but not for higher dimensions. The criterion is more effective than Bell's inequality in detecting inseparability.