Measure and detection of genuine multipartite entanglement for tripartite systems

It is a computationally hard task to certify genuine multipartite entanglement (GME). We investigate the relation between the norms of the correlation vectors and the detection of GME for tripartite quantum systems. A sufficient condition for GME and an effective lower bound for the GME concurrence are derived. Several examples are considered to show the effectiveness of the criterion and the lower bound of GME concurrence.

Figure 1

Vicente criterion (dashed line) vs the lower bound (5) in this paper (solid line). We have used $f\left(x\right)$ to denote the difference between the left and the right side of the two criteria. From the figure one sees that $\rho$ is genuine multipartite entangled for $0.894427<x\le 1$ by Vicente's result with $f\left(x\right)=-2.177324+2.434322x$. One computes $f\left(x\right)=-4.83138+6.74552x$ by our proposition. Thus GME is detected for $0.716235<x\le 1$.

Figure 2

Lower bound of GME concurrence in Theorem 2 for $\rho$ in Example 2. $g(x,y)$ stands for the lower bound.

Figure 3

GME detected by Vicente criterion (pink region by Theorem 1 and yellow region by Theorem 2 in [19]) and by the lower bound for GME concurrence in our Theorem 2 (blue region).

Figure 4

GME detection: dashed line by Theorem 1 in [19]; dotted and solid lines by our Theorem 1 and Theorem 2, respectively. $f\left(x\right)$ denotes the difference between the left and the right side of the inequalities in these criteria. We compute that $f\left(x\right)=2.17732(-1+x),-3.628874+4.044882x$ and $max\left\{0.222222\right(-3+3.4641x),0\}$ for Theorem 1 in [19], our Theorem 1 and Theorem 2, respectively. One finds from the figure that Theorem 1 in [19] cannot detect GME for the whole region of $x$ in this example, Theorem 2 in [19] cannot be operated on this state since it only fits for tripartite qubit systems, while Theorem 1 and Theorem 2 in this paper detect GME for $0.9<x\le 1$ and $0.866025<x\le 1$, respectively.