Characterizing the entanglement of bipartite quantum systems

Entangled states have been known almost from the very beginning of quantum mechanics @1,2#, and their somewhat unusual features have been investigated for many years. However, recent developments in the theory of quantum information @3# have required a deeper knowledge of their properties. The simplest system where one can study entanglement is represented by a bipartite system. In such a system, either with discrete or continuous variables, the inseparability of pure states is now well understood and the von Neumann entropy of either subsystem quantifies the amount of entanglement @4#. Instead, the question of inseparability of mixed states is much more complicated and involves subtle effects. For discrete variable systems the Peres-Horodecki criterion @5# constitutes a theoretical tool to investigate the separability. Recently, different criteria have been also proposed for continuous variable systems @6–11#. Nevertheless, a unifying criterion, of practical use, does not exist yet. Needless to say that also the entanglement quantification for mixed states is not well assessed @12#. On the other hand, the lack of knowledge for bipartite entanglement is not only a serious drawback in the study of mixed-state entanglement, but also a limitation for understanding multipartite entanglement. The aim of this paper is to throw some light on the plethora of entanglement criteria for bipartite systems. In particular, we shall derive a general separability criterion valid for any state of any bipartite system. We shall then discuss its relation with the already known criteria.