Reduction criterion for separability

We introduce a separability criterion based on the positive map $\Gamma :\overrightarrow{\rho }\left(\mathrm{Tr}\rho \right)-\rho ,$ where $\rho$ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of $\Gamma$ and the identity into a non-negative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the special case where one subsystem is a quantum bit, $\Gamma$ reduces to time reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\times 2$ and $2\times 3$ systems. Finally, a simple connection between this map for two qubits and complex conjugation in the “magic” basis [Phys. Rev. Lett. 78, 5022 (1997)] is displayed.