Separability Criterion for Density Matrices

A quantum system consisting of two subsystems is separable if its density matrix can be written as $\rho =\Sigma A^{}{w}_A{\rho }_A^{\prime }\otimes {\rho }_A^{\prime \prime },$ where ${\rho }_A^{\prime }$ and ${\rho }_A^{\prime \prime }$ are density matrices for the two subsystems, and the positive weights $w_A$ satisfy $\Sigma w_A=1\mathrm{}$. 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.