Characterization of entanglement of more than two qubits with Bell inequalities and global entanglement

It is shown that the entanglement-structure of 3- and $4\text{−}\text{qubit}$ states can be characterized by optimized operators of the Mermin-Klyshko type. It is possible to discriminate between pure $2\text{−}\text{qubit}$ entanglements and higher entanglements. A comparison with a global entanglement measure is made.

Figure 1

Optimization of the expectation value of $F_3$ for the $3\text{−}\text{qubit}$ generalized GHZ state as a function of $g^2$; with the following labeling: $\max \left\{\mid ⟨F_3⟩\mid \right\}$ (circles), $\mid ⟨F_3^{\prime }⟩\mid$ (squares), ${⟨F_3⟩}^2+{⟨F_3^{\prime }⟩}^2$ (triangles), and $\max \{{⟨F_3⟩}^2+{⟨F_3^{\prime }⟩}^2\}$ (down triangles); the course of $\max \left\{\mid ⟨F_3⟩\mid \right\}$ is fitted with $y=c_{0}g\sqrt{1-g^2}$, with $c_0=8.01$; the course of ${⟨F_3⟩}^2+{⟨F_3^{\prime }⟩}^2$ with $y=c_1{g}^2(1-g^2)$ and $c_1=64.01$; $\max \{{⟨F_3⟩}^2+{⟨F_3^{\prime }⟩}^2\}$ marks the optimization of the squared inequalities. The dashed lines mark the boundaries arising from the inequalities (3, 4). See text for more discussion.

Figure 2

Comparison of global entanglement $Q$ and the results of the $F_3$ optimization for the state $\mid {\psi }_5⟩$ (global entanglement $Q$: solid line and optimized $F_3$: circles) and the state $\mid {\psi }_7⟩$ (global entanglement $Q$: dashed line and optimized $F_3$: triangles); additionally the optimization of ${⟨F_3⟩}^2+{⟨F_3^{\prime }⟩}^2$ for the state $\mid {\psi }_7⟩$ is plotted (squares). The horizontal dashed line marks the transition point arising from the inequality (4).

Figure 3

Measurement of the superposition of $\mid {\psi }_7⟩$ and $\mid {\psi }_8⟩$; left $y$ axis: global entanglement (dashed line), sum of squared concurrences (solid line with triangles), and tangle (solid line); right $y$ axis: optimized $F_3$ operator (circles).

Figure 4

Concurrences, global entanglement, and $F_4$ optimization for the state $\mid {\Phi }_1⟩$; (a): $C_{ij}(J,J_s=2)$; (b): $C_{cj}(J_s,J=2)$; ($C_{12}$, $C_{23}$: dark solid line; $C_{13}$: dark dashed line; $C_{14}$, $C_{34}$: light solid line; $C_{24}$: light dashed line); (c) left $y$ axis: $Q$, right $y$ axis: $F_4$ optimization, as a function of $J$, $J_s=2$; and (d): left $y$ axis: $Q$, right $y$ axis: $F_4$ optimization, as a function of $J_s$, $J=2$ (global entanglement $Q$: solid line; optimized $F_4$: circles).

Figure 5

Comparison of the sum of the squared concurrences (solid line), one minus the sum of the squared concurrences (dashed line), and the $F_4$ optimization (circles) for the state $\mid {\Phi }_2⟩$.