Natural Multiparticle Entanglement in a Fermi Gas

We investigate multipartite entanglement in a noninteracting fermion gas, as a function of fermion separation, starting from the many particle fermion density matrix. We prove that all multiparticle entanglement can be built only out of two-fermion entanglement. Although from the Pauli exclusion principle we would always expect entanglement to decrease with fermion distance, we surprisingly find the opposite effect for certain fermion configurations. The von Neumann entropy is found to be proportional to the volume for a large number of particles even when they are arbitrarily close to each other. We will illustrate our results using different configurations of two, three, and four fermions at zero temperature although all our results can be applied to any temperature and any number of particles.

Figure 1

The negativity $N_{[2,13]}$ is plotted for fermion 2 moving from fermion 1 to fermion 3. This corresponds to the text values $i=1$, $j=3$, $k=2$. The distance between 1 and 3 is fixed to $x_{\max }=5$. This is a dimensionless number.

Figure 2

The negativities $N_{[2,13]}=N_{[3,12]}$ (solid line), $N_{[1,23]}$ for an isosceles triangle, for which fermions 2 and 3 form the base are plotted. Fermion 1 is moving away from the midpoint of the base. This corresponds to the text values $i=2$, $j=3$, and $k=1$.

Figure 3

The negativities $N_{[1,2]}$, $N_{[1,23]}$ (solid line) and $N_{[1,234]}$ (dashed line) are plotted. This corresponds to text values $p,q=1,2$, $l,mn=1,23$, and $i,jkl=1,234$.

Figure 4

The von Neumann entropy $S_2$, $S_3$ (solid line), $S_4$ (dashed line) is plotted as a function of fermion distance. This behavior is analytically explained in the text.