Existence criterion of genuine tripartite entanglement

In this paper, an intuitive mathematical formulation is provided to generalize the residual entanglement for tripartite systems of qubits [Phys. Rev. A 61, 052306 (2000)] to the tripartite systems in higher dimension. The spirit lies in the tensor treatment of tripartite pure states [Phys. Rev. A 72, 022333 (2005)]. A distinct characteristic of the present generalization is that the formulation for higher dimensional systems is invariant under permutation of the subsystems, hence is employed as a criterion to test the existence of genuine tripartite entanglement. Furthermore, the formulation for pure states can be conveniently extended to the case of mixed states by utilizing the Kronecker product approximate technique. As applications, we give the analytic approximation of the criterion for weakly mixed tripartite quantum states and consider the existence of genuine tripartite entanglement of some weakly mixed states.

Figure 1

The tensor grid of the coefficients of a tripartite pure state in $2\times 2\times 3$ dimensions.

Figure 2

$F_a\left(\rho \right)$ of the mixed state $\rho \left(x\right)=x\mid \mathrm{GHZ}⟩⟨\mathrm{GHZ}\mid +(1-x)∕2(\mid W⟩⟨W\mid +\mid \tilde{W}⟩⟨\tilde{W}\mid )$ vs $x$, $x∊(1∕3,1]$.