Entanglement criterion for tripartite systems based on local sum uncertainty relations

We propose a sufficient criterion for the entanglement of tripartite systems based on local sum uncertainty relations for arbitrarily chosen observables of subsystems. This criterion generalizes the tighter criterion for bipartite systems introduced by Zhang et al. [C.-J. Zhang, H. Nha, Y.-S. Zhang, and G.-C. Guo, Phys. Rev. A 81, 012324 (2010)] and can be used for both discrete- and continuous-variable systems. It enables us to detect the entanglement of quantum states without having a complete knowledge of them. Its utility is illustrated by some examples of three-qubit, qutrit-qutrit-qubit, and three-mode Gaussian states. It is found that, in comparison with other criteria, this criterion is able to detect some three-qubit bound entangled states more efficiently.

Figure 1

Plot of $F_{{\sigma }_b}^{AB|C}$ in terms of $b$.

Figure 2

Plot of $F_{{\varrho }_2}^{AB|C}$ in terms of $p$.

Figure 3

Plot of $F_{{\gamma }_a}^{AB|C}$ in terms of $a$ for two choices $a_1=0.5$, and $a_2=1$ (solid curve) and $a_1=0.5$ and $a_2=1.5$ (dashed curve).

Figure 4

Plot of $F_{{\gamma }_{\alpha }}^{AB|C}$ in terms of $\alpha$ for $a=1.2$.