Genuine tripartite entanglement in the noninteracting Fermi gas

We study genuine tripartite entanglement shared among the spins of three localized fermions in the noninteracting Fermi gas at zero temperature. First, we prove analytically with the aid of entanglement witnesses that in a particular configuration the three fermions are genuinely tripartite entangled in spin. Then various three-fermion configurations are investigated in order to quantify and calculate numerically the amount of genuine tripartite entanglement present in the system. Further we give a lower and an upper limit to the maximum diameter of the three-fermion configuration below which genuine tripartite entanglement exists and find that this distance is comparable with the maximum separation between two entangled fermions. The upper and lower limit turn out to be very close to each other indicating that the applied witness operator is well suited to reveal genuine tripartite entanglement in the collection of noninteracting fermions.

Figure 1

(Color online) Lower bound to the generalized robustness $E_R$ of genuine tripartite entanglement shared by three fermions is plotted for the 3D Fermi gas for the cases (a) fermion 2 is moved away from the position of fermion 1 toward the position of fermion 3, (b) fermion 2 is moved away from the midpoint of the line connecting fermion 1 and fermion 3 normal to this line (see the respective schematic geometrical pictures). The curves are plotted in both cases for the following dimensionless distances between fermion 1 and fermion 3: $k_{F}r\to 0$ (solid line) and $k_{F}r=0.5,1,1.5,2,2.5,2.59$ (dashed lines).

Figure 2

(Color online) Witnessed genuine tripartite entanglement among three fermions is plotted for the 3D Fermi gas for the configuration where fermion 2 [with polar coordinates $(\theta ,q)$] is allowed to move in a disk of radius $r∕2$ centered at the midpoint between fermion 1 and fermion 3, as shown by the geometrical picture. The polar plot represents curves that separate the regions associated with GTE (left-hand side) from regions without having GTE (right-hand side) for values of the dimensionless distances $k_{F}r\to 0$, $k_{F}r=1,2,2.5,2.59$, respectively.

Figure 3

(Color online) Lower bound to the generalized robustness $E_R$ of genuine tripartite entanglement is plotted for both the two- and three-dimensional degenerate Fermi gases in the function of the dimensionless distance $k_{F}r$, where $r$ is the relative distance between the positions of fermion 1 and fermion 3, and fermion 2 is located at the midpoint, as shown by the upper part of the figure.

Figure 4

(Color online). The three shaded half-disks represent areas corresponding to biseparable states with respect to the partitions $1\mid 23$, $12\mid 3$ and $13\mid 2$ on the $(r_1,r_2)$ plane by the section $r_{+}=0.041$ and $r_3=0$ for the 3D Fermi gas. The solid straight lines bound the set of biseparable states $B$ at this particular section. The equation $r_2=\sqrt{3}{r}_1$ is represented by the dashed line, and the cross stands for the point which is located on this line with coordinate $r_1=-0.35$.