Separability of Tripartite Quantum Systems

i pi = 1. Many separability criteria have been found in recent years. For pure states, the problem is completely solved, e.g., by using the Schmidt decomposition [1]. For mixed states, there are separability criteria such as PPT, reduction, majorization, realignment etc. [2, 3, 4, 5, 6]. In [7] the authors have given a lower bound of concurrence for tripartite quantum states which can be used to detect entanglement. In [8] the authors have provided a numerically computable criterion which can detect PPT entangled states for three qubits systems. The efficient criteron is then generalized to tripartite systems with arbitrary dimensions [9]. In [10] some nice results shew that some quantity related to Hermitian matrix is positive for quantum mixed states in 2×N systems, which was further discussed in [11]. These results were generalized to arbitrary dimensional bipartite systems (or 2×2×N quantum systems) in [12]. In this paper, we study arbitrary tripartite systems in analogue to the approach used in [10, 11, 12]. The properties of tripartite density matrices are studied in terms of the Bloch representations. A necessary condition for the separability of tripartite states has been obtained. These results are non-trivial when they are reduced to bipartite systems discussed in [10, 11, 12] and the separability criterion do detect some entanglements.