ua nt - p h / 03 08 15 9 v 1 2 8 A ug 2 00 3 Separability and entanglement in C 2 ⊗ C 3 ⊗ C N composite quantum systems

As one of the most striking features of quantum phenomena [1], quantum entanglement is playing very important roles in quantum information processing such as quantum computation [2], quantum teleportation [3, 4, 5, 6] (for experimental realization see [7]), dense coding [8] and quantum cryptographic schemes [9, 10, 11]. The separability of pure states for bipartite systems is quite well understood (cf. [12]). Nevertheless, The study of separability and entanglement of quantum mixed states is far from being satisfied. A mixed state is considered to be entangled if it is not a mixture of product states [13]. As the quantum correlations are weakened in mixed states, the manifestations of mixed-state entanglement can be very subtle [13, 14, 15]. To investigate the structure of mixed state entanglement some beautiful works have been done in quantifying entanglement [16, 17, 18, 19, 20, 21] for bipartite systems and multipartite systems (see e.g. [22, 23]). However most proposed measures of entanglement for bipartite systems involve extremizations which are difficult to handle analytically. For multipartite systems, one even does not know how to define the measures. Till now there is no general criterion that allows one to distinguish whether a mixed state is separable or not.