Separability Criterion for Density Matrices

A quantum system consisting of two subsystems is {ital separable} if its density matrix can be written as {rho}={summation}{sub {ital Aw}}{sub {ital A}}{rho}{prime}{sub {ital A}}{circle_times}{rho}{double_prime}{sub {ital A}}, where {rho}{prime}{sub {ital A}} and {rho}{double_prime}{sub {ital A}} are density matrices for the two subsystems, and the positive weights {ital w}{sub {ital A}} satisfy {summation}{ital w}{sub {ital A}}=1. In this Letter, it is proved that a necessary condition for separability is that a matrix, obtained by partial transposition of {rho}, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell{close_quote}s inequality for detecting quantum inseparability. {copyright} {ital 1996 The American Physical Society.}