respect to a bipartition $A|BC$, that is $\rho_{A|BC}^{sep}=\sum_{j}p_j|\varphi_A^j\rangle\langle\varphi_A^j|\otimes|\varphi_{BC}^j\rangle\langle\varphi_{BC}^j|$, where $p_j$ is a coefficient. Similarly, we denote other separable states for the two other bipartitions as $\rho_{B|AC}^{sep}$ and $\rho_{C|AB}^{sep}$. A biseparable state is a mixture of separable states, and combines the separable states $\rho_{A|BC}^{sep}$, $\rho_{B|AC}^{sep}$, and $\rho_{C|AB}^{sep}$ with respect to all possible bipartitions. Any state that is not a biseparable state is called genuinely multipartite entangled.

Recently, a powerful technique has been advanced to characterize multipartite entanglement using positive partial transpose (PPT) mixtures [@peres]. It is well known that a separable state is PPT, implying that its partial transpose is positive semidefinite. We denote a PPT mixture of a tripartite state as a convex combination of PPT states $\rho_{A|BC}^{PPT}$, $\rho_{B|AC}^{PPT}$ and $\rho_{C|AB}^{PPT}$ with respect to different bipartitions. The set of PPT mixtures contains the set of biseparable states. The advantage of using PPT mixtures instead of biseparable states is that the set of PPT mixtures can be fully characterized by the linear semidefinite programming (SDP) [@boyd], which is a standard problem of constrained convex optimization theory.

In order to characterize PPT mixtures, a multipartite state which is not a PPT mixture can be detected by a decomposable entanglement witness $W$ [@novo]. 