Separability Criterion for Density Matrices

A quantum system consisting of two subsystems is separable if its density matrix can be written as ρ = ΣAwAρA′⊗ρA′′, where ρA′ and ρA′′ are density matrices for the two subsystems, and the positive weights wA satisfy ΣwA = 1. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of ρ, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell's inequality for detecting quantum inseparability.