Hierarchy in temporal quantum correlations

A quantum system consisting of two subsystems is separable if its density matrix can be written as \(\rho=\sum_A w_A\,\rho_A'\otimes\rho_A''\), where \(\rho_A'\) and \(\rho_A''\) are density matrices for the two subsytems. In this Letter, it is shown that a necessary condition for separability is that a matrix, obtained by partial transposition of \(\rho\), has only non-negative eigenvalues. This criterion is stronger than Bell's inequality.

We provide necessary and sufficient conditions for separability of mixed states. As a result we obtain a simple criterion of separability for \(2\times2\) and \(2\times3\) systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. However, it is not the case in general. Some examples of mixtures which demonstrate the utility of the criterion are considered.