On Entanglement and Separability

We present a new necessary and sufficient condition to determine the entanglement status of an arbitrary N-qubit quantum state (maybe pure or mixed) represented by a density matrix. A necessary condition satisfied by separable bipartite quantum states was obtained by A. Peres, [1]. A. Peres showed that if a bipartite state represented by the density matrix is separable then its partial transpose is positive semidefinite and has no negative eigenvalues. In other words, if the partial transpose is not positive semidefinite and so one or more of its eigenvalues are negative then the state represented by the corresponding density matrix is entangled. It was then shown by M. Horodecki et.al, [2], that this necessary condition is also sufficient for two-by-two and two-by-three dimensional systems. However, in other dimensions, it was shown by P. Horodecki, [3], that the criterion due to A. Peres is not sufficient. In this paper, we develop a new approach and a new criterion for deciding the entanglement status of the states represented by the density matrices corresponding to N-qubit systems. We begin with a 2-qubit case and then show that these results for 2-qubit systems can be extended to N-qubit systems by proceeding along similar lines. We discuss few examples to illustrate the method proposed in this paper for testing the entanglement status of few density matrices.