The Separability versus Entanglement Problem

This chapter defines formally what entangled states are, presents some important criteria to discriminate entangled states from separable ones, and shows their classification according to their capability to perform some precisely defined tasks. Operational entanglement criteria for bipartite quantum states included are the partial transposition criterion, the majorization criterion, the cross-norm or realignment criterion, and the covariance matrix criterion. They detect whether a quantum state is separable or entangled. Some non-operational separability criteria introduced are not state-independent and provide important insights into the structure of the set of entangled states. The concept of entanglement witnesses can be used to detect entanglement experimentally, by performing only a few local measurements, assuming some prior knowledge of the density matrix. The chapter further shows how the Hahn-Banach theorem can be used to obtain “entanglement witnesses". It concludes by presenting a few remarks on multipartite states and multipartite entanglement.